The study of Martin‐Löf randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin‐Löf randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin‐Löf randomness may not be a natural notion because there is no universal test, and Martin‐Löf randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient (...) condition for the existence and coincidence, and study how much we can weaken this condition. (shrink)

The aim of this book is to give an account of Einstein's work without introducing anything very technical in the way of mathematics, physics, or philosophy.

The logical theory of branching space-times, which is intended to provide a framework for studying objective indeterminism, remains at a certain distance from the discussion of space-time theories in the philosophy of physics. In a welcome attempt to clarify the connection, Earman has recently found fault with the branching approach and suggested ``pruning some branches from branching space-time''. The present note identifies the different---order theoretic vs. topological---points of view of both discussion as a reason for certain misunderstandings, and tries to (...) remove them. Most importantly, we give a novel, topological criterion of modal consistency that usefully generalizes the order-theoretic criterion of directedness, and we introduce a differential-geometrical version of BST based on the theory of non-Hausdorff manifolds. (shrink)

The logical theory of branching space-times, which provides a relativistic framework for studying objective indeterminism, remains mostly disconnected from discussions of space-time theories in philosophy of physics. Earman has criticized the branching approach and suggested “pruning some branches from branching space-time.” This article identifies the different—order-theoretic versus topological—perspective of both discussions as a reason for certain misunderstandings and tries to remove them. Most important, we give a novel, topological criterion of modal consistency that usefully generalizes an earlier criterion, and we (...) introduce a differential-geometrical version of branching space-times as a non-Hausdorff (generalized) manifold. (shrink)

"The last remnant of physical objectivity of space-time" is disclosed, beyond the Leibniz equivalence, in the case of a continuous family of spatially non-compact models of general relativity. The physical individuation of point-events is furnished by the intrinsic degrees of freedom of the gravitational field, (viz, the "Dirac observables") that represent - as it were - the "ontic" part of the metric field. The physical role of the "epistemic" part (viz. the "gauge" variables) is likewise clarified. At the end, a (...) peculiar four-dimensional "holistic and structuralist" view of space-time emerges which includes elements common to the tradition of both substantivalism and relationism. The observables of our models undergo real "temporal change" and thereby provide a counter-example to the thesis of the "frozen-time" picture of evolution. (shrink)

This paper deals with a number of technical achievements that are instrumental for a dis-solution of the so-called "Hole Argument" in general relativity. Such achievements include: 1) the analysis of the "Hole" phenomenology in strict connection with the Hamiltonian treatment of the initial value problem. The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the "weak" ADM energy; 2) a re-interpretation of "active" diffeomorphisms as "passive and metric-dependent" (...) dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (up to now unknown) connection to gauge transformations on-shell; understanding such connection also enlightens the real content of the Hole Argument or, better, dis-solves it together with its alleged "indeterminism"; 3) the utilization of the Bergmann-Komar "intrinsic pseudo-coordinates", defined as suitable functionals of the Weyl curvature scalars, as tools for a peculiar gauge-fixing to the super-hamiltonian and super-momentum constraints; 4) the consequent construction of a "physical atlas" of 4-coordinate systems for the 4-dimensional "mathematical" manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent "Dirac observables"). Such construction embodies the "physical individuation" of the points of space-time as "point-events", independently of the presence of matter, and associates a "non-commutative structure" to each gauge fixing or four-dimensional coordinate system; 5) a clarification of the multiple definition given by Peter Bergmann of the concept of "(Bergmann) observable" in general relativity. This clarification leads to the proposal of a "main conjecture" asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). A by-product of this achievements is the falsification of a recently advanced argument asserting the absence of (any kind of) "change" in the observable quantities of general relativity. 6) a clarification of the physical role of Dirac and gauge variables as their being related to "tidal-like" and "inertial-like" effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define notion of "force" in general relativity in a natural way; 7) a proposal showing how the physical individuation of point-events could in principle be implemented as an experimental setup and protocol leading to a "standard of space-time" more or less like atomic clocks define standards of time. We conclude that, besides being operationally essential for building measuring apparatuses for the gravitational field, the role of matter in the non-vacuum gravitational case is also that of "participating directly" in the individuation process, being involved in the determination of the Dirac observables. This circumstance leads naturally to a peculiar new kind of "structuralist" view of the general-relativistic concept of space-time, a view that embodies some elements of both the traditional "absolutist" and "relational" conceptions. In the end, space-time point-events maintain a "peculiar sort of objectivity". Some hints following from our approach for the quantum gravity programme are also given. (shrink)

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial (...) time. Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

The application of the general theory of relativity to the cosmological plane has led to a significant change in traditional notions on the space-time structure of the universe. This has been expressed in a new formulation and solution of cosmological problems such as that of a single world space and time, the problem of selection of a cosmological model for description of the universe, and the problem of the infinity of the universe.

A generalization of the familiar de Broglie-Bohm interpretation of quantum mechanics is formulated, based on relinquishing the momentum relationship p=∇S and allowing a spread of momentum values at each position. The development of this framework also provides a new perspective on the well-known question of joint distributions for quantum mechanics. It is shown that, for an extension of the original model to be physically acceptable and consistent with experiment, it is necessary to impose certain restrictions on the associated joint distribution (...) for particle positions and momenta. These requirements thereby define a new class of possible models. In pursuing this line of reasoning, the main contributions of this paper are (i) to identify the restrictions that must be imposed, (ii) to demonstrate that joint distribution expressions satisfying them do exist, and (iii) to construct a sample model based on one such joint distribution. (shrink)

The history of the theory of general relativity presents unique features. After its discovery, the theory was immediately confirmed and rapidly changed established notions of space and time. The further implications of general relativity, however, remained largely unexplored until the mid 1950s, when it came into focus as a physical theory and gradually returned to the mainstream of physics. This essay presents a historiographical framework for assessing the history of general relativity by taking into account in an integrated narrative intellectual (...) developments, epistemological problems, and technological advances; the characteristics of post–World War II and Cold War science; and newly emerging institutional settings. It argues that such a framework can help us understand this renaissance of general relativity as a result of two main factors: the recognition of the untapped potential of general relativity and an explicit effort at community building, which allowed this formerly disparate and dispersed field to benefit from the postwar changes in the scientific landscape. (shrink)

A view is put forward, according to which various aspects of the structure of the world as internalized by the brain take the form of “neural spaces,” a concrete counterpart for Shepard's “abstract” ones. Neural spaces may help us understand better both the representational substrate of cognition and the processes that operate on it. [Shepard].

A survey is presented of the essential principles for formulating relativistic wave equations in curved spacetime. The approach is relatively simple and avoids much of the philosophical debate about covariance principles, which is also indicated. Hypercomplex numbers provide a natural language for covariance symmetry and the two important kinds of covariant derivative.

This volume offers an integrated understanding of how the theory of general relativity gained momentum after Einstein had formulated it in 1915. Chapters focus on the early reception of the theory in physics and philosophy and on the systematic questions that emerged shortly after Einstein's momentous discovery. They are written by physicists, historians of science, and philosophers, and were originally presented at the conference titled Thinking About Space and Time: 100 Years of Applying and Interpreting General Relativity, held at the (...) University of Bern from September 12-14, 2017. By establishing the historical context first, and then moving into more philosophical chapters, this volume will provide readers with a more complete understanding of early applications of general relativity and of related philosophical issues. Because the chapters are often cross-disciplinary, they cover a wide variety of topics related to the general theory of relativity. These include: Heuristics used in the discovery of general relativity Mach's Principle The structure of Einstein's theory Cosmology and the Einstein world Stability of cosmological models The metaphysical nature of spacetime The relationship between spacetime and dynamics The Geodesic Principle Symmetries Thinking About Space and Time will be a valuable resource for historians of science and philosophers who seek a deeper knowledge of the uses of general relativity, as well as for physicists and mathematicians interested in exploring the wider historical and philosophical context of Einstein's theory. (shrink)

In this paper, we introduce the concept of N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. This is the first of a series of papers devoted to the investigation of the Killing symmetries of generalized Minkowski spaces. In particular, we discuss here the infinitesimal-algebraic structure of the space-time rotations in such spaces. It is shown that the maximal Killing group of these spaces is the (...) direct product of a generalized Lorentz group and a generalized translation group. We derive the explicit form of the generators of the generalized Lorentz group in the self-representation and their related, generalized Lorentz algebra. The results obtained are specialized to the case of a 4-dimensional, “deformed” Minkowski space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\widetilde M}$$ \end{document}4, i.e., a pseudoeuclidean space with metric coefficients depending on energy. (shrink)

In this paper, we continue the study of the Killing symmetries of an N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space–time rotations in such spaces, by confining ourselves to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space M_4, for which we derive the explicit general form of (...) the finite rotations and boosts in different parametric bases. (shrink)

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the translations in such spaces, by confining ourselves to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widetilde M_4 $$ \end{document}.

We describe the (enormous) size of the search space for Dynamic Casual Models and generalizations of them.

We use the theory of triposes to prove that every locale H is the set of truth values of a complete and co-complete quasi-topos into which the category of topological spaces embeds and the topos of sheaves over H reflectively embeds.

Techniques for constructing the tensor product of two generalized sample spaces which admit unital sets of dispersion-free weights are discussed. A duality theory is developed, based on the 1-cuts of the dispersion-free weights, and used to produce a candidate for the tensor product. This construction is verified for Dacification manuals, a conjecture is given for other reflexive cases, and some adjustments for nonreflexive cases are considered. An alternate approach, using graphs of interpretation morphisms on the duals, is also presented.

The paper focuses on some puzzles about Carnap's intended epistemological point in the "completion" and "generalization" of the Anschauungsraum in sec. II of Der Raum (leaving aside the technical problems which also arise). Since any global structure at all requires that eidetic intuition be supplemented with freely-chosen postulates and/or intuitively unmotivated generalizations, it is unclear, as several authors have pointed out, how and in what sense "intuitive space" as a whole represents a distinctive, a priori contribution to our knowledge. I (...) suggest a way of approaching this issue based on Carnap's sources -- in particular, Husserl and Driesch, both of whom he repeatedly claims to be following. The idea of a severely finite realm of possible intuition, which both requires and allows supplementation with an infinite conceptual structure, is central to Husserl's thought, and, I argue, it would be natural for Carnap to rely on it in attempting to reconcile Husserlian eidetic intuition with the general theory of relativity. That this larger conceptual structure owes its details to free postulation is, on the other hand, decidedly un-Husserlian. But here, I claim, Carnap takes his cue from Driesch's view, in the first edition of the Ordnungslehre, that natural actuality is the result of a certain demand for order: we demand such order in natural things and reject as non-actual (hallucinatory, dreamed, etc.) whatever nature-like elements of experience fail to fit into it. In conclusion I suggest that more radical versions of this particular constellation of ideas provide the key to understanding much of Carnap's later thought. (shrink)

In this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver‐null (...) sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems. (shrink)

We start by giving a survey to the theory of $${\text {Borel}}^{*}$$ sets in the generalized Baire space $${\text {Baire}}=\kappa ^{\kappa }$$. In particular we look at the relation of this complexity class to other complexity classes which we denote by $${\text {Borel}}$$, $${\Delta _1^1}$$ and $${\Sigma _1^1}$$ and the connections between $${\text {Borel}}^*$$ sets and the infinitely deep language $$M_{\kappa ^+\kappa }$$. In the end of the paper we will prove the consistency of $${\text {Borel}}^{*}\ne \Sigma ^{1}_{1}$$.

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains with (...) Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, _L_-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions. (shrink)

An analysis of some of the applications of Clifford space relativity to the physics behind the modified black hole entropy-area relations, rainbow metrics, generalized dispersion and minimal length stringy uncertainty relations is presented.

Given an uncountable regular cardinal κ, we study the structural properties of the class of all sets of functions from κ to κ that are definable over the structure 〈H,∈〉 by a Σ1-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of ZFC together with large cardinal axioms. In this paper, we present other canonical extensions of ZFC that provide a strong structure theory for these classes. These axioms are (...) variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins. (shrink)

The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacuum state. As an introduction into the subject, a toy model of the harmonic oscillator in pseudo-Euclidean space is (...) studied. (shrink)

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators βμ (p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases are (...) obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators. (shrink)

In this paper, we introduce the notion of S ∗ p -partial metric spaces which is a generalization of S-metric spaces and partial-metric spaces. Also, we give some of the topological properties that are important in knowing the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorems in this spaces.

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the (...) Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations. (shrink)

The current technoscientific progress has led to a sectorization in the philosophy of science. Today the philosophy of science isn't is informal interested in studying old problems about the general characteristics of scientific practice. The interest of the philosopher of science is the study of concepts, problems and riddles of particular disciplines. Then, within this progress of philosophy of science, neuroscientific research stands out, because it invades issues traditionally addressed by the humanities, such as the nature of consciousness, action, knowledge, (...) normativity, etc. As a result, the new area of interdisciplinary study of neuroscience and philosophy arises: neurophilosophy. This emerging area applies neuroscientific concepts to traditional philosophical questions, limiting their responses to neuroscientific revelations about nervous systems. Neurophilosophy research focuses on problems related to the indirect nature of mind and brain, computational or representative analysis of brain process, relationships between psychological and neuroscientific research, adequate adaptations of physical and philosophical concepts in neuroscience and the place of cognitive functions. Now, the temporal representation of conscious experience and the types of the neural architecture to represent objects in time have aroused scientific interest. Under these interests, we focus on the studies on the temporary triadic structure of phenomenological consciousness in Dan Lloyd and Rick Grush. From Grush’s studies, the importance of Kantian ideas for cognitive neuroscience emerges, due to the active way in which Kant conceived space and time as forms of intuition, within which the mind interprets its experience. Under this perspective, the theoretical arguments of Dennett-Kinsbourne and Eagleman-Sejnowski represent winks in the direction of Kant-Husserl within the neuroscientific goal while considering that the contents provided by the mind included space, objects and perception of causal relationships. Then, theories of cognitive neuroscience are beginning to suggest that these elements are also, as Kant argued, interpretative elaborations provided by mind / brain, and not only content received from outside. In other words, current cognitive neuroscientific theories try to pass from its Humean phase to a Kantian phase. So, the challenge has been to explain that these elements are provided by the mind and the world itself, and how they have the content they have come from. These are lacking in current studies. Filling this gap helps to involve the analysis of the scientist’s experience in his theoretical attitude. In this sense, an investigation under the Kantian-Husserlian approach that involves pure intuitions a priori with the experience of the scientific and neuroscientific concepts represents a ground-breaking. At present, a neurophilosophical study about this does not exist. In this sense, one feasible proposal for research would be based on the application of neuroscientific results of Moser-Britt to philosophical problems of foundational notions in relativistic physics: space, time, space-time, field, etc., under the Kantian-Husserlian approach, which allows to demonstrate the multidisciplinary link between neurophilosophy and physics. This represents a ground-breaking area in current interests in scientific research, with a positive impact in the field of neuroscience, and contributing to the study of abstraction emphasizing the importance of Kant’s Copernican turn and Husserl’s phenomenological ideas in the construction of physical theories. -/- . (shrink)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

In this book John Campbell shows that the general structural features of human thought can be seen as having their source in the distinctive ways in which we...

In 1906, Frigyes Riesz introduced a preliminary version of the notion of a topological space. He called it a mathematical continuum. This development can be traced back to the end of 1904 when, genuinely interested in taking up Hilbert’s foundations of geometry from 1902, Riesz aimed to extend Hilbert’s notion of a two-dimensional manifold to the three-dimensional case. Starting with the plane as an abstract point-set, Hilbert had postulated the existence of a system of neighbourhoods, thereby introducing the notion of (...) an accumulation point for the point-sets of the plane. Inspired by Hilbert’s technical approach, as well as by recent developments in analysis and point-set topology in France, Riesz defined the concept of a mathematical continuum as an abstract set provided with a notion of an accumulation point. In addition, he developed further elementary concepts in abstract point-set topology. Taking an abstract topological approach, he formulated a concept of three-dimensional continuous space that resembles the modern concept of a three-dimensional topological manifold. In 1908, Riesz presented his concept of mathematical continuum at the International Congress of Mathematicians in Rome. His lecture immediately won the attention of people interested in carrying on his research. They promoted his ideas, thus assuring their gradual reception by several future founders of general topology. In this way, Riesz’s work contributed significantly to the emergence of this discipline. (shrink)

I suggest that the difficulties inherent in discovering the hidden regularities in realistic (type-2) problems can often be resolved by learning algorithms employing simple constraints (such as symmetry and the importance of local information) that are natural from an evolutionary point of view. Neither “heavy-duty nativism” nor “representational recoding” appear to offer totally appropriate descriptions of such natural learning processes.

Mental Spaces is the classic introduction to the study of mental spaces and conceptual projection, as revealed through the structure and use of language. It examines in detail the dynamic construction of connected domains as discourse unfolds. The discovery of mental space organization has modified our conception of language and thought: powerful and uniform accounts of superficially disparate phenomena have become available in the areas of reference, presupposition projection, counterfactual and analogical reasoning, metaphor and metonymy, and time and aspect in (...) discourse. The present work lays the foundation for this research. It uncovers simple and general principles that lie behind the awesome complexity of everyday logic. (shrink)

We propose an alternative resolution of Simpson's paradox in multiple classification experiments, using a different maximum likelihood estimator. In the center of our analysis is a formal representation of free choice and randomization that is based on the notion of incompatible measurements.We first introduce a representation of incompatible measurements as a collection of sets of outcomes. This leads to a natural generalization of Kolmogoroff's axioms of probability. We then discuss the existence and uniqueness of the maximum likelihood estimator for a (...) probability weight on such a generalized sample space, given absolute frequency data.As a first example, we discuss an estimation problem with censured data that classically admits only biased ad hoc estimators.Next, we derive an explicit solution of the maximum likelihood estimation problem for a large class of experiments that arise from various kids of compositions of sample spaces. We identify the (categorical) direct sum of sample spaces as a representation of “free choice,” and the (categorical) direct product as a representation of “randomization.”Finally, we apply the foregoing discussion to the case of multiple classification experiments in order to show that there is no Simpson's paradox if the difference between free choice and randomization is recognized in the structure of the experiment.A comparison between our new estimator and the “usual” calculation can be summarized as follows: Pooling the data over one classification factor in the “usual” way in fact destroys or ignores the information contained in it, whereas our proposed maximum likelihood estimator is a proper marginal over this factor that “averages out” the information contained in it. The estimators agree with each other in the case of proportional sample sizes. (shrink)