A timely collection of advanced, original material in the area of statistical methodology motivated by geometric problems, dedicated to the influential work of Kanti V. Mardia This volume celebrates Kanti V. Mardia's long and influential career in statistics. A common theme unifying much of Mardia’s work is the importance of geometry in statistics, and to highlight the areas emphasized in his research this book brings together 16 contributions from high-profile researchers in the field. Geometry Driven Statistics covers a wide range (...) of application areas including directional data, shape analysis, spatial data, climate science, fingerprints, image analysis, computer vision and bioinformatics. The book will appeal to statisticians and others with an interest in data motivated by geometric considerations. Summarizing the state of the art, examining some new developments and presenting a vision for the future, Geometry Driven Statistics will enable the reader to broaden knowledge of important research areas in statistics and gain a new appreciation of the work and influence of Kanti V. Mardia. (shrink)

The stochastic phase-space solution of the particle localizability problem in relativistic quantum mechanics is reviewed. It leads to relativistically covariant probability measures that give rise to covariant and conserved probability currents. The resulting particle propagators are used in the formulation of stochastic geometries underlying a concept of quantum spacetime that is operationally based on stochastically extended quantum test particles. The epistemological implications of the intrinsic stochasticity of such quantum spacetime frameworks for microcausality, the EPR paradox, etc., are discussed.

The action-reaction principle (AR) is examined in three contexts: (1) the inertial-gravitational interaction between a particle and space-time geometry, (2) protective observation of an extended wave function of a single particle, and (3) the causal-stochastic or Bohm interpretation of quantum mechanics. A new criterion of reality is formulated using the AR principle. This criterion implies that the wave function of a single particle is real and justifies in the Bohm interpretation the dual ontology of the particle and its associated (...) wave function. But it is concluded that the Bohm theory is not dynamically complete because the particle and its associated wave function do not satisfy the AR principle. (shrink)

The roles of classical realism, logical positivism, and pragmatic instrumentalism in the shaping of fundamental ideas in quantum physics are examined in the light of some recent historical and sociological studies of the factors that influenced their development. It is shown that those studies indicate that the conventionalistic form of instrumentalism that has dominated all the major post-World War II developments in quantum physics is not an outgrowth of the Copenhagen school, and that despite the “schism” in twentieth century physics (...) created by the Bohr-Einstein “disagreements” on foundational issues in quantum theory, both their philosophical stands were very much opposed to those of conventionalistic instrumentalism. Quotations from the writings of Dirac, Heisenberg, Popper, Russell, and other influential thinkers, are provided, illustrating the fact that, despite the various divergencies in their opinions, they all either opposed the instrumentalist concept of “truth” in general, or its conventionalistic version in post-World War II quantum physics in particular. The basic epistemic ideas of a quantum geometry approach to quantum physics are reviewed and discussed from the point of view of a quantum realism that seeks to reconcile Bohr's “positivism” with Einstein's “realism” by emphasizing the existence of an underlying quantum reality, in which they both believed. This quantum geometry framework seeks to introduce geometro-stochastic concepts that are specifically designed for the systematic description of that underlying quantum reality, by developing the conceptual and mathematical tools that are most appropriate for such a use. (shrink)

in Stochastic Processes, Physics and Geometry II, edited by S. Albeverio, U. Cattaneo, D. Merlini (World Scientific, Singapore, 1995) pp. 221-232.

Part I Introduction -/- Passion at a Distance (Don Howard) -/- Part II Philosophy, Methodology and History -/- Balancing Necessity and Fallibilism: Charles Sanders Peirce on the Status of Mathematics and its Intersection with the Inquiry into Nature (Ronald Anderson) -/- Newton’s Methodology (William Harper) -/- Whitehead’s Philosophy and Quantum Mechanics (QM): A Tribute to Abner Shimony (Shimon Malin) -/- Bohr and the Photon (John Stachel) -/- Part III Bell’s Theorem and Nonlocality A. Theory -/- Extending the Concept of an (...) “Element of Reality” to Work with Inefficient Detectors (Daniel M. Greenberger) -/- A General Proof of Nonlocality without Inequalities for Bipartite States (GianCarlo Ghirardi and Luca Marinatto) -/- On the Separability of Physical Systems (Jon P. Jarrett) -/- Bell Inequalities: Many Questions, a Few Answers (Nicolas Gisin) -/- B. Experiment -/- Do Experimental Violations of Bell Inequalities Require a Nonlocal Interpretation of Quantum Mechanics? II: Analysis a la Bell (Edward S. Fry, Xinmei Qu, and Marlan O. Scully) -/- The Physics of 2 = 1 + 1 (Yanhua Shih) -/- Part IV Probability, Uncertainty, and Stochastic Modifications of Quantum Mechanics -/- Interpretations of Probability in Quantum Mechanics: A Case of “Experimental Metaphysics” (Geoffrey Hellman) -/- “No Information Without Disturbance”: Quantum Limitations of Measurement (Paul Busch) -/- How Stands Collapse II (Philip Pearle) -/- Is There a Relation Between the Breakdown of the Superposition Principle and an Indeterminacy in the Structure of the Einsteinian Space-Time? (Andor Frenkel) -/- Indistinguishability or Stochastic Dependence? (D. Costantini and U. Garibaldi) -/- Part V Relativity -/- Plane Geometry in Spacetime (N. David Mermin) -/- The Transient nows (Steven F. Savitt) -/- Quantum in Gravity? (Michael Horne) -/- A Proposed Test of the Local Causality of Spacetime (Adrian Kent) -/- Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure (Lucien Hardy) -/- “Definability,” “Conventionality,” and Simultaneity in Einstein–Minkowski Space-Time (Howard Stein) -/- Part VI Concluding Words -/- Bistro Banter: A Dialogue with Abner Shimony and Lee Smolin -/- Unfinished Work: A Bequest (Abner Shimony) -/- Bibliography of Abner Shimony. (shrink)

Special relativity is most naturally formulated as a theory of spacetime geometry, but within the spacetime framework probability appears to be a purely epistemic notion. It is possible that progress can be made with rather different approaches - covariant stochastic equations, in particular - but the results to date are not encouraging. However, it seems a non-epistemic notion of probability can be made out in Minkowski space on Everett's terms. I shall work throughout with the consistent histories formalism. I (...) shall start with a conservative interpretation, and then go on to Everett's. (shrink)

The strong correlation between the geometry of the dendritic tree and the specific function of motoneurons suggests that their synaptic contacts are established on a selective stochastic basis with the characteristic form of dendrites being the source of selection in the frog. A compromise is suggested according to which specific structures may have evolved on a selective stochastic basis and “constructive learning” could be the source of selection in the cortex.

The role of CPT invariance and consequences for bipartite entanglement of neutral (K) mesons are discussed. A relaxation of CPT leads to a modification of the entanglement which is known as the ω effect. The relaxation of assumptions required to prove the CPT theorem are examined within the context of models of space-time foam. It is shown that the evasion of the EPR type entanglement implied by CPT (which is connected with spin statistics) is rather elusive. Relaxation of locality (through (...) non-commutative geometry) or the introduction of an environment do not by themselves lead to a destruction of the entanglement. One model of the environment, which is based on non-critical strings and D-particle capture and recoil, leads to a specific momentum dependent stochastic contribution to the space-time metric and consequent change in the neutral meson bipartite entanglement. Although the class of models producing the omega effect is non-empty, the lack of an omega effect is demonstrated for a wide class of models based on thermal like baths which are often considered as generic models appropriate for the study of space-time foam. (shrink)

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes equations for (...) viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations. (shrink)

This paper considers the nature of time and temporality in Homer. It argues that any exploration of narrative and time must, as its central tenet, take into account the irreducible plurality and interconnectedness of memory, the event, and experienced time. Drawing on notions of complexity, emergence, and stochastic behavior in science as well as phenomenological traditions in the discussion and analysis of time, temporality, and change, and offering extensive readings of Homer, of Homeric epithets and formulae, and of key (...) passages in the Iliad and Odyssey, the paper argues against chronological notions of linear time and progression and in favor of a complex, dynamic temporal “geometries” of Homeric temporality. The paper concludes by briefly extending the argument to the wider domain of ancient time in general. Homer is a fundamental point of reference in the ancient world. Thus, Homeric temporality—irreducibly complex—affects the cognition and perception of time throughout the whole of antiquity. (shrink)

We report the results of an optical analogue of the fullerene molecule diffraction experiment. Our results, and an analysis of the fullerene experiment, suggest that the patterns observed in the latter can be explained using a localized particle model. There is no evidence that the grating period contributed to the published fullerene diffraction pattern. De Broglie waves, if they exist, are unlikely to have played a significant part in the fullerene diffraction experiment. The observed patterns are not consistent with those (...) expected according to wave theory for the experimental geometry corresponding to the slit-detector system and the de Broglie wavelength. The measurements were performed in the near field, making the demonstration of wave properties difficult. We outline a new classical approach to the electron and neutron interference experiments. The magnetic moment is crucial to this model, which emphasizes a mechanism for generating narrow-band continuum X-radiation. Some experiments are proposed which can decide between the suggested model and quantum mechanics, and which can also rule out an alternative stochastic model. (shrink)

In the traditional formalism of quantum mechanics, a simple direct proof of the Spin Geometry Theorem of Penrose is given; and the structure of a model of the ‘space of the quantum directions’, defined in terms of elementary SU-invariant observables of the quantum mechanical systems, is sketched.

Disoriented animals from ants to humans reorient in accord with the shape of the surrounding surface layout: a behavioral pattern long taken as evidence for sensitivity to layout geometry. Recent computational models suggest, however, that the reorientation process may not depend on geometrical analyses but instead on the matching of brightness contours in 2D images of the environment. Here we test this suggestion by investigating young children's reorientation in enclosed environments. Children reoriented by extremely subtle geometric properties of the 3D (...) layout: bumps and ridges that protruded only slightly off the floor, producing edges with low contrast. Moreover, children failed to reorient by prominent brightness contours in continuous layouts with no distinctive 3D structure. The findings provide evidence that geometric layout representations support children's reorientation. (shrink)

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the proponent of (...) the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

ABSTRACT A spatial logic is a modal logic of which the models are the mathematical models of space. Successively considering the mathematical models of space that are the incidence geometry and the projective geometry, we will successively establish the language, the semantical basis, the axiomatical presentation, the proof of the decidability and the proof of the completeness of INC, the modal multilogic of incidence geometry, and PRO, the modal multilogic of projective geometry.

This article introduces Harvey Brown and Oliver Pooley’s ‘dynamical approach’ to special relativity, and argues that it may be construed as a relationalist form of Einstein’s ‘practical geometry’. This construal of the dynamical approach is shown to be compatible with related chapters of Brown’s text and also with recent descriptions of the dynamical approach by Pooley and others.

A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system (...) are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results. (shrink)

The goal of this study was to implement a Riemannian geometry -based algorithm to detect high mental workload and mental fatigue using task-induced electroencephalogram signals. In order to elicit high MWL and MF, the participants performed a cognitively demanding task in the form of the letter n-back task. We analyzed the time-varying characteristics of the EEG band power features in the theta and alpha frequency band at different task conditions and cortical areas by employing a RG-based framework. MWL and MF (...) were considered as too high, when the Riemannian distances of the task-run EEG reached or surpassed the threshold of the baseline EEG. The results of this study showed a BP increase in the theta and alpha frequency bands with increasing experiment duration, indicating elevated MWL and MF that impedes/hinders the task performance of the participants. High MWL and MF was detected in 8 out of 20 participants. The Riemannian distances also showed a steady increase toward the threshold with increasing experiment duration, with the most detections occurring toward the end of the experiment. To support our findings, subjective ratings and behavioral measures were also considered. (shrink)

According to Kant’s philosophy of geometry, Euclidean geometry is synthetic _a priori_. The advent of non-Euclidean geometries proved this position at least problematic, if not obsolete. However, based on Nash’s embedding theorems we show that a weaker notion of _preeminence_ supports the view that Euclidean geometry, even though not strictly _a priori_, enjoys a more fundamental status than non-Euclidean geometries.

A survey of Kant's views on space, time, geometry and the synthetic nature of mathematics. I concentrate mostly on geometry, but comment briefly on the syntheticity of logic and arithmetic as well. I believe the view of many that Kant's system denied the possibility of non-Euclidean geometries is clearly mistaken, as Kant himself used a non-Euclidean geometry (spherical geometry, used in his day for navigational purposes) in order to explain his idea, which amounts to an anticipation of the later discovery (...) of the general concept of non- Euclidean geometries. Kant's view of geometry and arithmetic as synthetic was, I believe, essentially correct, in that geometry and arithmetic are both synthetic a priori if considered as branches of mathematics independent of the rest of mathematics. However, the view that somehow logic is analytic, while mathematics is synthetic for Kantian reasons, is mistaken. All three disciplines—logic, arithmetic and geometry—are synthetic as disciplines independent from one another. However, they have a common basis, recursion theory, which I prefer to identify with mathematics as a whole. As a result, I do not say, as is often considered to be the Kantian view, that mathematics is synthetic while logic is analytic. Rather, I prefer to say that mathematics is analytic, while logic is synthetic. This is perfectly consistent with Kant's system, since it was arithmetic and geometry individually that he argued were synthetic. What Kant called the analytic is recursion theory, which could be considered as a basic formulation of mathematics or logic—or better, both mathematics and logic could be recognized as essentially the same discipline. However, if "logic" is taken to mean "predicate logic", as is often the case in modern times, then it is mathematics that is closer to Kant's analytic, not logic. Such ambiguities, of course, can be avoided by simply associating Kant's analytic with recursion theory, and avoiding the controversies as to what counts as mathematics or logic.. (shrink)

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he draws between a precise and (...) an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)

The iterated Prisoner’s Dilemma has become the standard model for the evolution of cooperative behavior within a community of egoistic agents, frequently cited for implications in both sociology and biology. Due primarily to the work of Axelrod (1980a, 198Ob, 1984, 1985), a strategy of tit for tat (TFT) has established a reputation as being particularly robust. Nowak and Sigmund (1992) have shown, however, that in a world of stochastic error or imperfect communication, it is not TFT that finally triumphs (...) in an ecological model based on population percentages (Axelrod and Hamilton 1981), but ‘generous tit for tat’ (GTFT), which repays cooperation with a probability of cooperation approaching 1 but forgives defection with a probability of l/3. In this paper, we consider a spatialized instantiation of the stochastic Prisoner’s Dilemma, using two-dimensional cellular automata (Wolfram, 1984, 1986; Gutowitz, 1990) to model the spatial dynamics of populations of competing strategies. The surprising result is that in the spatial model it is not GIFT but still more generous strategies that are favored. The optimal strategy within this spatial ecology appears to be a form of ‘bending over backwards’, which returns cooperation for defection with a probability of 2/3 - a rate twice as generous as GTFT. (shrink)

Do evolutionary processes such as selection and random drift cause evolutionary change, or are they merely convenient ways of describing or summarizing it? Philosophers have lined up on both sides of this question. One recent defense (Reisman and Forber 2005) of the causal status of selection and drift appeals to a manipulability theory of causation. Yet, even if one accepts manipulability, there are still reasons to doubt that genetic drift, in particular, is genuinely causal. We will address two challenges to (...) treating drift as causal within a manipulation framework. We will argue that both challenges ultimately fail, but that they raise interesting and subtle issues about the nature of causation and the differences between selection and drift. ‡Thanks to audiences at the ‘PBDB1' Conference and the 2006 PSA Meeting for valuable feedback. †To contact the authors, please write to: Patrick Forber, Tufts University, Philosophy Department, Miner Hall, Medford, MA 02155; e-mail: [email protected]. (shrink)

We argue that current constructive approaches to the special theory of relativity do not derive the geometrical Minkowski structure from the dynamics but rather assume it. We further argue that in current physics there can be no dynamical derivation of primitive geometrical notions such as length. By this we believe we continue an argument initiated by Einstein.

This autodidactic paper wraps up an earlier epistemic art project and compactly collates the main unfolded scientific and philosophical strategies for epistemic resiliency against epistemic doom in the deepfake era. Retrospectively speaking, the existence of a dense condensate within which explanatory blockchain (EB) based science, EB-based philosophy and EB-based art overlap acts as a pointer to untapped non-algorithmic epistemic resources that could (if ever activated) exhibit the natural tendency to compel the reach of algorithmic computations – noticeably at the "cost" (...) of increasing non-algorithmic self-determination. (shrink)

The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams (...) are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

Sunto Prendendo in esame quanto il celebre maestro di retorica Marco Fabio Quintiliano (35 d.C. ca - 100 d.C. ca) scrive in età flavia nella sua _Institutio oratoria_ a proposito dell’importanza dello studio della Matematica nella formazione di base del futuro perfetto oratore romano, si intende approfondire in particolare una porzione del lungo passo presente nel I libro (I 10, 34-49), nello specifico i §§ 39-45. In essi l’autore latino, partendo dall’affermazione che la geometria, non meno dell’aritmetica, con il suo (...) procedimento razionale smaschera ciò che, pur apparendo verisimile, in realtà è falso, inserisce in funzione esemplificativa una vera e propria lezione di geometria piana sul problema dell’isoperimetria. Lo scopo dell’autore è di fornire una prova dell’utilità della geometria per educare a un uso corretto dell’_ars_ _dicendi_. _Keywords__: _Quintiliano; retorica; isoperimetria. Abstract Examining what the famous rhetoric teacher Marcus Fabius Quintilian (ca. 35 CE – ca. 100 CE) writes during the Flavian period in his Institutio Oratoria about the importance of studying mathematics as part of the foundational education for the future perfect Roman orator, this paper focuses on a specific section of the long passage found in Book I (I 10, 34-49), particularly §§ 39-45. In this section, the Latin author, starting from the assertion that geometry, no less than arithmetic, with its rational method, reveals what may appear plausible but is actually false, includes a true lesson in plane geometry on the problem of isoperimetry. The author's goal is to demonstrate the utility of geometry in educating one to make proper use of the ars dicendi (art of speaking). _Keywords: _ Quintilian; rhetoric; isoperimetry. (shrink)

In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept (...) crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs. (shrink)

This book offers a general introduction to the geometrical studies of Gottfried Wilhelm Leibniz and his mathematical epistemology. In particular, it focuses on his theory of parallel lines and his attempts to prove the famous Parallel Postulate. Furthermore it explains the role that Leibniz’s work played in the development of non-Euclidean geometry. The first part is an overview of his epistemology of geometry and a few of his geometrical findings, which puts them in the context of the 17th-century studies on (...) the foundations of geometry. It also provides a detailed mathematical and philosophical commentary on his writings on the theory of parallels, and discusses how they were received in the 18th century as well as their relevance for the non-Euclidean revolution in mathematics. The second part offers a collection of Leibniz’s essays on the theory of parallels and an English translation of them. While a few of these papers have already been published (in Latin) in the standard Leibniz editions, most of them are transcribed from Leibniz’s manuscripts written in Hannover, and published here for the first time. The book provides new material on the history of non-Euclidean geometry, stressing the previously neglected role of Leibniz in these developments. (shrink)

The relationship between physics and geometry is examined in classical and quantum physics based on the view that the symmetry group of physics and the automorphism group of the geometry are the same. Examination of quantum phenomena reveals that the space-time manifold is not appropriate for quantum theory. A different conception of geometry for quantum theory on the group manifold, which may be an arbitrary Lie group, is proposed. This provides a unified description of gravity and gauge fields as well (...) as generalizations of these fields. A correspondence principle which relates the geometry of quantum physics and the geometry of classical physics is formulated. (shrink)

Philosophy in the period immediately after Aristotle is sometimes thought to be marked by the decline of natural philosophy and philosophical disinterest in contemporary achievements in the sciences. But in one area at least, the early third century B.C.E. was a time of productive interaction between such disparate fields as epistemology, physics and geometry. Debates between the sceptics and the dogmatic philosophical schools focus on epistemological problems about the possibility of self-evident appearances, but there is evidence from Euclid's day of (...) a quite different response. The sceptical challenge provoked the development of theories explaining error formation, showing how illusions can be studied systematically and are subject to prediction. Such theories do not legitimate claims about the nature of the underlying entities perceived, but provide justification for forming expectations about future perceptions. While it overtly focuses on purely geometrical considerations, the Euclidean model of optics nonetheless provides support for certain views about the nature of vision and the physics of light. Moreover, by offering a model in which the image received is not thought to be a perspicuous mirroring of the object seen, Euclid may have helped promote a view of perception as something reconstructed from information received, not as a mere form transferred into the eye. The ancient sceptic may indeed have fulfilled his promise to promote inquiry by focusing attention on problems that escape the attention of a hasty theorist. (shrink)

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory.