One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question of whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is (...) yes: I state and sketch proofs of two technical results—inspired by simple physical arguments about the generic properties of classical systems—to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none provided by Hamiltonian. The argument not only clarifies the conceptual structure of the two systems of mechanics, but also their relations to each other and their respective mechanisms for representing physical systems. It also shows why naïvely structural approaches to the representational content of physical theories cannot work. [Lagrange] grasped that he had gained a method of stating dynamical truths in a way, which is perfectly indifferent to the particular methods of measurement employed in fixing the positions of the various parts of the system. Accordingly, he went on to deduce equations of motion, which are equally applicable whatever quantitative measurements have been made, provided that they are adequate to fix positions. The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things. (Whitehead [1948], p. 63)1. Introduction2.Classical Systems3. The Possible Interactions of a Classical System and the Structure of Its Space of States4. Classical Systems Are Lagrangian5. Classical Systems Are Not Hamiltonian6. How Lagrangian and Hamiltonian Mechanics Represent Classical Systems7. The Conceptual Structure of Classical Mechanics. (shrink)

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive `equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories provides a sense in which they are isomorphic, and mutually and canonically definable through an analog of `common definitional extension'.

One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is yes: (...) I state and prove two technical results, inspired by simple physical arguments about the generic properties of classical systems, to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none that Hamiltonian mechanics does. The argument not only clarifies the conceptual structure of the two systems of mechanics, their relations to each other, and their respective mechanisms for representing physical systems. It also provides a decisive counter-example to the semantical view of physical theories, and one, moreover, that shows its crucial deficiency: a theory must be, or at least be founded on, more than its collection of models (in the sense of Tarski), for a complete semantics requires that one take account of global structures defined by relations among the individual models. The example also shows why naively structural accounts of theory cannot work: simple isomorphism of theoretical and empirical structures is not rich enough a relation to ground a semantics. (shrink)

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted furthermore as (...) the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined. (shrink)

What is the proper metaphysics of quantum mechanics? In this dissertation, I approach the question from three different but related angles. First, I suggest that the quantum state can be understood intrinsically as relations holding among regions in ordinary space-time, from which we can recover the wave function uniquely up to an equivalence class (by representation and uniqueness theorems). The intrinsic account eliminates certain conventional elements (e.g. overall phase) in the representation of the quantum state. It also dispenses with first-order (...) quantification over mathematical objects, which goes some way towards making the quantum world safe for a nominalistic metaphysics suggested in Field (1980, 2016). Second, I argue that the fundamental space of the quantum world is the low-dimensional physical space and not the high-dimensional space isomorphic to the ``configuration space.'' My arguments are based on considerations about dynamics, empirical adequacy, and symmetries of the quantum mechanics. Third, I show that, when we consider quantum mechanics in a time-asymmetric universe (with a large entropy gradient), we obtain new theoretical and conceptual possibilities. In such a model, we can use the low-entropy boundary condition known as the Past Hypothesis (Albert, 2000) to pin down a natural initial quantum state of the universe. However, the universal quantum state is not a pure state but a mixed state, represented by a density matrix that is the normalized projection onto the Past Hypothesis subspace. This particular choice has interesting consequences for Humean supervenience, statistical mechanical probabilities, and theoretical unity. (shrink)

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, (...) neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (shrink)

A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincare transformations is introduced. For a class of functions in H that are well localized in the time variable the usual formalism of non-relativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in H becomes (...) the usual Shrodinger evolution with t as a parameter. The relativistic invariance of the construction is proved. The usual theory of relativity on Minkowski space-time is shown to be ``isometrically and equivariantly embedded'' into H. That is, classical space-time is isometrically embedded into H, Poincare transformations have unique extensions to isomorphisms of H and the embedding commutes with Poincare transformations. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

An axiomatic theory is formulated which describes a class of “yes-no” experiments, involving a fixed basic source, a fixed basic detector, and various filters. It is assumed that all filters considered can be constructed from a setP of primitive filters by composition and stochastic selection. Two physically plausible axioms are formulated which allow us to define the concept of asystem in the present context (cf. Definition2.4). To each system we can attach anorder unit module ( $^\circ \hat V, ^\circ \hat (...) V_ + , |1\rangle , \langle 1|$ ) (cf. Definition5.1) whereby ( $^\circ \hat V, ^\circ \hat V_ + , |1\rangle ,$ ) is acomplete, separable order unit space. Two additional axioms are proposed which have the effect that the space ( $^\circ \hat V, ^\circ \hat V_ + , |1\rangle ,$ ) becomes isomorphic to the order unit space underlying a JB-algebra, at least in the case where $^\circ \hat V$ isfinite dimensional (cf. Corollary7.9). (shrink)

This article presents a three-stage model of how isomorphic mechanisms have shaped corporate social responsibility reporting practices over time. In the first stage, defensive reporting, companies fail to meet stakeholder expectations due to a deficiency in firm performance. In this stage, the decision to report is driven by coercive isomorphism as firms sense pressure to close the expectational gap. In the second stage, proactive reporting, knowledge of CSR reporting spreads and the practice of CSR reporting becomes normatively sanctioned. (...) In this stage, normative isomorphism leads other organizations to look to CSR reporting as a potential new opportunity for achieving the firm’s goals. In the third stage, imitative diffusion, the defensive reporters together with the proactive reporters create a critical mass of CSR reporters that reaches a threshold at which the benefits of CSR reporting begin to outweigh any costs due to mimetic isomorphism. The study finds support for the model in an examination of Fortune 500 firms from 1997 to 2006. (shrink)

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and re-presentable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, (...) and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen. (shrink)

The purpose of this research is to provide an overview of state governance for corporate social responsibility (CSR) in the countries of the Gulf Cooperation Council (GCC). A systemic literature review method is employed to collect 88 relevant publications, and a qualitative coding method is used to identify 98 governance instruments from those publications. These are grouped into 13 themes and then examined within three conceptual models. The findings reveal that most of the instruments are geared towards ethical expectations, internal (...) and external social responsibility, raising awareness, and socio‐economic development. The findings are then explored within four theories. The results suggest that CSR governance in the GCC is at a moderate to high level of bureaucratization; reflects Islamic identity, national development targets, and business accountability; is between the stages of habitualization and objectification; and relies mostly on normative pressures. Recommendations for policymakers and company managers are then presented based on these findings. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

A formal model of metaphor is introduced. It models metaphor, first, as an interaction of “frames” according to the frame semantics, and then, as a wave function in Hilbert space. The practical way for a probability distribution and a corresponding wave function to be assigned to a given metaphor in a given language is considered. A series of formal definitions is deduced from this for: “representation”, “reality”, “language”, “ontology”, etc. All are based on Hilbert space. A few statements about a (...) quantum computer are implied: The sodefined reality is inherent and internal to it. It can report a result only “metaphorically”. It will demolish transmitting the result “literally”, i.e. absolutely exactly. A new and different formal definition of metaphor is introduced as a few entangled wave functions corresponding to different “signs” in different language formally defined as above. The change of frames as the change from the one to the other formal definition of metaphor is interpreted as a formal definition of thought. Four areas of cognition are unified as different but isomorphic interpretations of the mathematical model based on Hilbert space. These are: quantum mechanics, frame semantics, formal semantics by means of quantum computer, and the theory of metaphor in linguistics. (shrink)

As universities are increasingly called by their national governments for a more entrepreneurial management of public research results, they started to develop internal structures and policies to take a proactive role in the commercialisation of university research. For the first time, this paper presents a detailed chronicle of how country-level reforms on Intellectual Property Rights were translated into organisation-level mechanisms to regulate university-patenting activity. The analysis is based on the complete list of patent policies issued between 1993 and 2009 (...) by the population of Italian universities. Our evidence suggests that universities first dealt with legislative changes on IPRs by enacting isomorphic behaviours, then by creating a community of practices, and finally by leveraging on such community to influence government reforms on IP-related matters. We discuss our results in the light of institutional theory and public policy. (shrink)

Recent work on the hole argument in general relativity by Weatherall has drawn attention to the neglected concept of models’ representational capacities. I argue for several theses about the structure of these capacities, including that they should be understood not as many-to-one relations from models to the world, but in general as many-to-many relations constrained by the models’ isomorphisms. I then compare these ideas with a recent argument by Belot for the claim that some isometries “generate new possibilities” in general (...) relativity. Philosophical orthodoxy, by contrast, denies this. Properly understanding the role of representational capacities, I argue, reveals how Belot’s rejection of orthodoxy does not go far enough, and makes better sense of our practices in theorizing about spacetime. (shrink)

According to recent accounts of the imagination, mental mechanisms that can take input from both imagining and from believing will process imagination‐based inputs (pretense representations) and isomorphic beliefs in much the same way. That is, such a mechanism should produce similar outputs whether its input is the belief that p or the pretense representation that p. Unfortunately, there seem to be clear counterexamples to this hypothesis, for in many cases, imagining that p and believing that p have quite (...) different psychological consequences. This paper sets out some central problem cases and argues that the cases might be accommodated by adverting to the role of desires concerning real and imaginary situations. (shrink)

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the quantum potential. This (...) equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium ; gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation. (shrink)

This article develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR, the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature—a theory's ‘not caring which point, or particle, is which’—supported a structuralist ontology. Pooley criticizes this analogy: (...) in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of particle symmetry, in which states (vectors, rays, or density operators) are not fixed by all permutations (called ‘paraparticle states’). Thus Stachel's analogy is revived. (shrink)

This position paper presents the beginnings of a general theory of representations starting from the notion that an intelligent agent is essentially a control system with multiple control states, many of which contain information (both factual and non-factual), albeit not necessarily in a propositional form. The paper attempts to give a general characterisation of the notion of the syntax of an information store, in terms of types of variation the relevant mechanisms can cope with. Similarly concepts of semantics pragmatics (...) and inference are generalised to apply to information-bearing sub-states in control systems. A number of common but incorrect notions about representation are criticised (such as that pictures are in some way isomorphic with what they represent). (shrink)

The abstract description of a physical system is developed, along lines originally suggested by Birkhoff and von Neumann, in terms of the complete lattice of propositions associated with that system, and the distinction between classical and quantum systems is made precise. With the help of the notion of state, a propositional system is defined: it is remarked that every irreducible propositional system (of more than three dimensions) is isomorphic to the lattice of all closed subspaces of a Hilbert space (...) constructed on some division ring with involution. The propositional system consisting of a family of separable complex Hilbert spaces is treated as a particular case which is sufficiently general to include both classical and quantum mechanics. The theory of the Galilean particle without spin is given as an illustration. Finally, the basis for the statistical interpretation of wave mechanics is developed with the help of Gleason's theorem. In an appendix, a proof of essentially the first part of Gleason's theorem is given which is a little different (perhaps more geometric) from that originally given by Gleason. (shrink)

Quantum probability is used to provide a new organization of basic quantum theory in a logical, axiomatic way. The principal thesis is that there is one fundamental time evolution equation in quantum theory, and this is given by a new version of Born’s Rule, which now includes both consecutive and conditional probability as it must, since science is based on correlations. A major modification of one of the standard axioms of quantum theory allows the implementation of various mathematically distinct models (...) (commonly called ‘pictures’) of them. A natural definition of isomorphism of models is presented next. For a given Hamiltonian the standard ‘pictures’ of quantum theory (e.g., Schrödinger, Heisenberg, interaction) are isomorphic models, whose probabilities are nonetheless model independent. The Schrödinger equation remains valid in one model of the axioms, even though it is no longer the fundamental equation of time evolution. All the usual calculations of standard, textbook quantum theory remain valid, but they are put in a new light. For example, the ‘collapse’ of the state is now viewed in the Schrödinger model as _one_ step in a _two_ step algorithm for calculating _one_ conditional probability. In the isomorphic Heisenberg model, where states are constant in time, one has the same conditional probability given by the same formula. Also, entanglement is defined in a new, more general model independent way as an aspect of quantum probability without using ‘collapse’ or tensor products. (shrink)

Under the influence of groundbreaking work by John Meyer and Brian Rowen, as well as Paul DiMaggio and Walter Powell, over the last 30 years research in the new sociological institutionalism has focused on processes of isomorphism. I argue that this is a one-sided focus that leaves out many insights from other institutional and macrosociological approaches and does not do justice to actual social change because it overlooks the role played by divergent institutional development. While the suggestion of divergent trends (...) is not new, there have been few attempts to integrate divergence into the theoretical premises of the new sociological institutionalism. Based on the typology proposed by DiMaggio and Powell, I show that the mechanisms identified by them as sources of isomorphic change can support processes of divergent change as well. The theoretical challenge is to identify conditions under which these mechanisms push institutional change toward homogenization or divergence. (shrink)

The persistent challenge of formulating ontic structuralism in a rigorous manner, which prioritizes structures over the entities they contain, calls for a transformation of traditional logical frameworks. I argue that Univalent Foundations (UF), which feature the axiom that all isomorphic structures are identical, offer such a foundation and are more attractive than other proposed structuralist frameworks. Furthermore, I delve into the significance in the case of the hole argument and, very briefly, the nature of symmetries.

Panmicropsychism is the view that the fundamental physical ingredients of our universe are also its fundamental phenomenal ingredients. Since there is only a limited number of fundamental physical ingredients, panmicropsychism seems to imply that there exists only a small set (palette) of basic phenomenal qualities. How does this limited palette of basic phenomenal qualities give rise to our rich set of experiences? This is known as ‘the palette problem’. One class of solutions to this problem, large-palette solutions, simply denies that (...) the palette is limited. These solutions assume that all types of phenomenal qualities (color, sound, odor, taste, etc., and presumably also types not experienced by humans) were created fully formed at the birth of our universe. On this view, brains evoke conscious experiences by sampling primordial, preexisting phenomenal spaces. My main claim in this paper is that, by analogy with the mathematical description of the fundamental physical ingredients of our universe, which exhibits simplicity, symmetry, and beauty, panmicropsychists should expect the mathematical description of the fundamental phenomenal ingredients of our universe to exhibit similar features. The goal of this paper is to exemplify this claim using what is arguably the simplest of all types of phenomenal qualities—color. Specifically, I utilize phenomenological data on color to construct the maximally symmetric mathematical description of phenomenal color space. I then show that this mathematical description is isomorphic to the mathematical description of two-state quantum systems in a mixed state. Based on this isomorphism, I suggest that color may be the phenomenal dual aspect of two-state quantum systems. (shrink)

The automorphism groups of scalar products of spinors are determined. Spinors are considered as elements of minimal left ideals of Clifford algebras on quadratic modules, e.g., on orthogonal spaces. Orthogonal spaces of any dimension and arbitrary signature are discussed. For example, the automorphism groups of scalar products of Pauli spinors and Dirac spinors are, respectively, isomorphic to the matrix groups U(2) and U(2, 2). It is found that there are, in general, 32 different types or similarity classes of such (...) automorphism groups, if one considers real orthogonal spaces of arbitrary dimension and arbitrary signature. On a more abstract level this means that the Brauer-Wall group of the real field R, consisting of graded central simple algebras over R, can be extended from the cyclic group of 8 elements to an abelian group with32 elements. This extended Brauer-Wall group is seen to be isomorphic with the group (Z 8 × Z 8 )/Z 2 and it consists of real Clifford algebras with antiinvolutions. Moreover, the subgroup determined by the antiinvolution is isomorphic to the automorphism group of scalar products of spinors. (shrink)

There has been an upsurge of interest lately in developing Wigner’s hypothesis that conscious observation causes collapse by exploring dynamical collapse models in which some purportedly quantifiable aspect of consciousness resist superposition. Kremnizer–Ranchin, Chalmers–McQueen and Okon–Sebastián have explored the idea that collapse may be associated with a numerical measure of consciousness. More recently, Chalmers–McQueen have argued that any single measure is inadequate because it will allow superpositions of distinct states of equal consciousness measure to persist. They suggest a satisfactory model (...) needs to associate collapse with a set of measures quantifying aspects of consciousness, such as the “Q-shapes” defined by Tononi et al. in their “integrated information theory” of consciousness. I argue here that Chalmers–McQueen’s argument against associating a single measure with collapse requires a precise symmetry between brain states associated with different experiences and thus does not apply to the only case where we have strong intuitions, namely human observers. In defence of Chalmers–McQueen’s stance, it might be argued that idealized artificial information processing networks could display such symmetries. However, I argue that the most natural form of any theory that postulates a map from network states to mind states is one that assigns identical mind states to isomorphic network states. This suggests that, if such a map exists, no familiar components of mind states, such as viewing different colours, or experiencing pleasure or pain, are likely to be related by symmetries. (shrink)

In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and corresponds to (...) a generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field (E≠0) into a pure electric field, but never into a pure magnetic field. (shrink)

Recent cognitive accounts of the imagination propose that imagining and believing are in the same “code”. According to the single code hypothesis, cognitive mechanisms that can take input from both imagining and from believing will process imagination-based inputs (“pretense representations”) and isomorphic beliefs in much the same way. In this paper, I argue that the single code hypothesis provides a unified and independently motivated explanation for a wide range of puzzles surrounding fiction.

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that (...) the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935). (shrink)

The susceptibility of decision-makers’ choices to variations in option framing has been attributed to individual differences in cognitive style. According to this view, individuals who are prone to a more deliberate, or less intuitive, thinking style are less susceptible to framing manipulations. Research findings on the topic, however, have tended to yield small effects, with several studies also being limited in inferential value by methodological drawbacks. We report two experiments that examined the value of several cognitive-style variables, including measures of (...) cognitive reflection, subjective numeracy, actively open-minded thinking, need for cognition, and hemispheric dominance, in predicting participants’ frame-consistent choices. Our experiments used an isomorph of the Asian Disease Problem and we manipulated frames between participants. We controlled for participants’ sex and age, and we manipulated the order in which choice options were presented to participants. In Experiment 1 (N = 190) using an undergraduate sample and in Experiment 2 (N = 316) using a sample of Amazon Mechanical Turk workers, we found no significant effect of any of the cognitive style measures taken on predicting frame-consistent choice, regardless of whether we analyzed participants’ binary choices or their choices weighted by the extent to which participants preferred their chosen option over the non-chosen option. The sole factor that significantly predicted frame consistent choice was framing: in both experiments, participants were more likely to make frame consistent choices when the frame was positive than when it was negative, consistent with the tendency towards risk aversion in the task. The present findings do not support the view that individual differences in people’s susceptibility to framing manipulations can be substantially accounted for by individual differences in cognitive style. (shrink)

In 1925 the old quantum mechanics of Planck, Einstein, and Bohr was replaced by the new (matrix) quantum mechanics of Born, Heisenberg, Jordan, and Dirac. In 1926 Schrödinger developed wave mechanics, which proved to be equivalent to matrix mechanics in the sense that they led to the same energy levels. Dirac and Jordan joined the two theories into one transformation quantum theory. In 1932 von Neumann presented his Hilbert space formulation of quantum mechanics and proved a representation theorem showing that (...) sequences in transformation theory were isomorphic to sequences in the Hilbert space formulation (see quantum mechanics). Three different notions of theory identity are involved here: theory individuation, theoretical equivalence, and empirical equivalence. (shrink)

Once the notion of a precursive relationship between developmental stages is fully articulated in terms of the distinction between ‘role’ and ‘realiser’ states, it turns out that the ‘Theory of Mind’ literature operates with a notion of precursive relationships described at too high a level of abstraction to explain actual mechanisms of development. Furthermore, the tendency within that literature to explain precursive relationships in terms of role states with isomorphic linguistic/computational structures is misleading. Developmental relationships are more likely (...) to exist between states which play a similar normative role in the agents’s psychology than between states which can be described as sharing a similar computational architecture. (shrink)

A corollary of Gleason's theorem asserts that if the lattice of propositions of a physical system is isomorphic to the lattice of subspaces of a Hilbert space of dimension greater than two, then there is no probability measure that assigns only the values 1 and 0 (truth and falsity, respectively) to each of the propositions. Belinfante outlined an elegant geometrical proof of this corollary but relied upon an unrigorous measure-theoretical statement. An amplified geometrical proof is given along Belinfante's lines, (...) but dispensing with measure theory. (shrink)

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...) to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, a fact (...) first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...) other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

Information processing theories of memory and skills can be reformulated in terms of how categories are physically and temporally related, a process called conceptual coordination. Dreaming can then be understood as a story-understanding process in which two mechanisms found in everyday comprehension are missing: conceiving sequences (chunking categories in time as a higher-order categorization) and coordinating across modalities (e.g., relating the sound of a word and the image of its meaning). On this basis, we can readily identify isomorphisms between (...) dream phenomenology and neurophysiology, and explain the function of dreaming as facilitating future coordination of sequential, cross-modal categorization (i.e., REM sleep lowers activation thresholds, “unlearning”). [Hobson et al.; Nielsen; Solms; Revonsuo; Vertes & Eastman]. (shrink)

We identify new, rather serious, physical and mathematical inconsistencies of the current formulation of noncanonical or nonunitary string theories due to the lack of invariant units necessary for consistent measurements, lack of preservation in time of Hermiticity-observability, and other shortcomings. We propose three novel reformulations of string theories for matter of progressively increasing complexity via the novel iso-, geno-, and hyper-mathematics of hadronic mechanics, which resolve the current inconsistencies, while offering new intriguing possibilities, such as: an axiomatically consistent and invariant (...) inclusion of gravity, the reduction of macroscopic irreversibility to the most primitive level of vibrations of the universal substratum (ether), or the treatment of multi-valued, irreducible, biological structures. We then identify three corresponding classical formulations of string theories for antimatter via the novel anti-isomorphic isodual mathematics. We finally outline the intriguing features of the emerging new cosmologies (including biological structures, as it should be for all cosmologies), such as: universal invariance (rather than covariance) under a symmetry isomorphic to the Poincaré group and its isodual; equal distributions of matter and antimatter in the universe (as a limit case); continuous creation; no need for the missing mass; significantly reduced dimensions; possibility of experimental identification of antimatter in the universe; identically null total characteristics of time, energy, linear and angular momentum, charge, etc.; and other intriguing features. (shrink)

The principles of classical physics, including deterministic dynamics and observability of physical states, are incompatible with the existence of unobservable conscious minds that possess free will. Attempts to directly accommodate consciousness in a classical world lead to philosophical paradoxes such as causally ineffective consciousness and possibility of alternate worlds in which functional brain isomorphs behave identically but lack conscious experiences. Here, we show that because Chalmers’ principle of organizational invariance is based on a deficient nineteenth century classical physics, it is (...) inherently flawed and implies evolutionary inexplicable epiphenomenal consciousness. Consequently, if consciousness is a fundamental ingredient of physical reality, no psychophysical laws such as Chalmers’ principle of organizational invariance are needed to establish correspondence between conscious experiences and brain function. Quantum mechanics is the most successful and only modern physical theory capable of naturally accommodating consciousness without violation of physical laws. (shrink)

I consider the direct product algebra formed from two isomorphic Clifford algebras. More specifically, for an element x in each of the two component algebras I consider elements in the direct product space with the form x ⊗ x. I show how this construction can be used to model the algebraic structure of particular vector spaces with metric, to describe the relationship between wavefunction and observable in examples from quantum mechanics, and to express the relationship between the electromagnetic field (...) tensor and the stress-energy tensor in electromagnetism. To enable this analysis I introduce a particular decomposition of the direct product algebra. (shrink)

Quantum stochastic models are developed within the framework of a measure entity. An entity is a structure that describes the tests and states of a physical system. A measure entity endows each test with a measure and equips certain sets of states as measurable spaces. A stochastic model consists of measurable realvalued function on the set of states, called a generalized action, together with measures on the measurable state spaces. This structure is then employed to compute quantum probabilities of test (...) outcomes. We characterize those measure entities that are isomorphic to a quantum probability space. We also show that stochastic models provide a phase space description of quantum mechanics and a realistic model of spin. (shrink)

What is often called “multiple” modernities is best seen as referring to the different expressions of an increasingly emergent global modernity rather than simply to multiple societal forms. Modernity is not converging into a unitary, homogenous form; rather it denotes an isomorphic condition of common aspirations, learning mechanisms, visions of the world, modes of communication. As such modernity can arise anywhere in the world; it is not a specific tradition or societal form but a mode of processing, or (...) translating, culture. Modernity is a particular way of transmitting culture that transforms that which it takes over; it is not a culture of its own and therefore can take root anywhere at any time. This is because every translation is a transformation of both the object and the subject. The essence of modernity is a capacity to transform culture in a continuous process of translation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Seeking the Everyday Meaning of Autonomy in Neurologic DisordersGeorge J. Agich (bio)The Socratic aphorism that the unexamined life is not worth living and dictums like "Know thyself" remind us of the centrality of self-understanding in the history of philosophical reflections on autonomy. These traditional concerns with autonomy may seem far removed from the neurologic impairments to which Joel Anderson and Warren Lux draw our attention. Nonetheless, Anderson and Lux (...) have provided an important discussion that links the traditional philosophical commitment to self-knowledge with an account that parses these concepts in the context of neurologic disorders. Theirs is a potentially productive approach for improving our understanding of autonomy, one that Emilio Mordini and I advocated in a paper entitled, "Autonomy and the Ethics of Neurosurgery" (Agich and Mordini 1998, 54). We argued that the focus of bioethics on issues such as informed consent misses the more challenging and potentially fruitful collaboration that neurosurgery and neurology affords for advancing the philosophical understanding of the conditions of autonomy. The challenge is in the integration of concepts used in one context of meaning with other domains of discourse. A good example of the challenges embedded in pursuing this kind of project can be found in Anderson and Lux's discussion of the concept of accurate self-assessment.Anderson and Lux argue that accurate self-assessment is a requirement for one to act autonomously. On first hearing, this phrase sounds unsurprising and consistent with standard approaches that assume a higher degree of awareness of one's capacities. Their characterization, however, derives from their observations of cases involving severe neurologic impairments, which point to more basic processes than the reflective awareness or knowledge that is often assumed. Paradoxically, the language of accurate self-assessment sounds remarkably like an intellectual function. It appears to fall squarely within the traditional philosophical tendency to define autonomy in terms of ego-centered, intellectual functions involving relatively high-level reflective capacities. Despite their terminology, Anderson and Lux stress that their account requires a type of reflexivity that is not the same as self-conscious reflection understood in the highest degree. They stress this in a number of places. For example, they see the neurologic concept of executive function as involving basic feedback mechanisms that are "broadly isomorphic with capacities associated [End Page 295] with autonomy;" these include anticipation, goal seeking, planning, initiation, sequencing, monitoring, error detection, self-correction, as well as initiation of novel responses, a clearly heterogenous set of capacities (Anderson and Lux 2004, 285). These capacities are not capacities of the "mind," but basic neurologic functions of the intact embodied agent who is actively engaged in the world. Thus, these concepts capture or express a practical (or clinical) as opposed to a theoretical understanding of the basic conditions or capacities defining autonomy. Similarly, their talk of "self-assessment" does not need to be understood as explicit or conscious; rather, it typically operates in the background (Anderson and Lux 2004). They term this character of self-assessment "mundane" (Anderson and Lux 2004, 289), which strikes me as exactly the right way to characterize this key feature of autonomy (Agich 1995). Their phenomenological and clinically based insights are quite sound, but the reader struggles with a terminology that seems to run in a different direction. To be sure, they stress that they intend their language of "autonomy" and "accurate self-assessment" to involve ordinary, background, and everyday functions that do not require high levels or significant capacities for reflective self-assessment, but their very insistence underscores the problem inherent in this terminology.Anderson and Lux argue that the capacity for self-assessment has a task-relative dimension that requires a degree of "accuracy." They characterize accuracy in a way that may promote rather than forestall misunderstanding. They state that "an adequate degree of precision" (281) is required in such assessment. For example, in assessing whether one is able to jump a hurdle, they say "one must accurately estimate not only the strength of one's legs but also the height of the hurdle" (Anderson and Lux 2004, 281). They further characterize their account of "accuracy" as involving an epistemological externalism, which might lead some readers to assume... (shrink)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...) the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

This paper contains a proposal for a free, nonzero-rest-mass particle’s proper spacetime, determined exclusively by the particle’s rest mass \ and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers \ that count de Broglie time units \\). The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by (...) the Compton unit \\). The spatial and temporal coordinates that form the particle’s proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term \, of an additional energy term of the same order of magnitude, which is related to large intervals of the \-particle’s proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed. (shrink)

In sensory neuroscience, the neural and perceptual levels of investigation are commonly related through a reductive research strategy based in psycho-neural isomorphisms. Davida Teller’s “linking propositions” are a particularly vivid illustration of this epistemology in the context of vision science. For Teller, linking propositions guide the core epistemological practices of vision science by expressing the criteria for acceptable explanations of perceptual phenomena by neural processes and by articulating heuristics for discovering neural properties on grounds of perceptual ones, and vice versa. (...) Furthermore, linking propositions create a uniform discourse in which ontological identity statements can be formulated on the basis of structural terms applicable to both the neurophysiological and perceptual domains.I argue that the epistemic practices of current color vision science are better understood through Machamer-style mechanisms than isomorphism-based linking propositions. For Machamer, activity types of mechanisms are central for both explanation and the discovery of entities and their properties. The activity type of color cancellation is fundamental to the opponent process paradigm of color vision precisely in this manner. The descriptive inadequacy of linking propositions for clarifying vision scientific research is particularly damaging because activity types have a unifying role for the structural facts of color perception. As a form of local reductionism, Teller’s proffered epistemology aims to create a uniform domain for neuroscience and perceptual psychology but on the basis of a non-unified, disparate set of structural identity statements. By bringing attention to the unifying role of activity types, Machamer mechanisms allow also for critical epistemological explorations of ontological reductions on grounds of the fit between activity types of higher-level and lower-level sciences. They offer a Machamer-ontological-critical or MOCing framework, for short, against local reductionism. (shrink)

Analogical transfer in sequence learning is presented as an example of how the type-2 problem of learning an unbounded number of isomorphic sequences is reduced to the type-1 problem of learning a small finite set of sequences. The commentary illustrates how the difficult problem of appropriate analogical filter creation and selection is addressed while avoiding the trap of strong nativism, and it provides theoretical and experimental evidence for the existence of dissociable mechanisms for type-1 learning and type-2 recoding.

Once the notion of a precursive relationship between developmental stages is fully articulated in terms of the distinction between ‘role’ and ‘realiser’ states, it turns out that the ‘Theory of Mind’ literature operates with a notion of precursive relationships described at too high a level of abstraction to explain actual mechanisms of development. Furthermore, the tendency within that literature to explain precursive relationships in terms of role states with isomorphic linguistic/computational structures is misleading. Developmental relationships are more likely (...) to exist between states which play a similar normative role in the agents’s psychology than between states which can be described as sharing a similar computational architecture. (shrink)

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.