This paper examines the contribution of Gabrio Piola to continuum mechanics.Though he was undoubtably a skilled mathematician and a good mechanician, little is commonly known about his papers within the international scientific community, principally because a large part of the Italian school of mechanics was isolated in the first half of the XIXth century.We examine and comment on Piola’s most important papers, and compare them with those of his contemporaries Cauchy, Poisson and Kirchhoff.

Many philosophers have claimed that there is a tension between the impenetrability of matter and the possibility of contact between continuous bodies. This tension has led some to claim that impenetrable continuous bodies could not ever be in contact, and it has led others to posit certain structural features to continuous bodies that they believe would resolve the tension. Unfortunately, such philosophical discussions rarely borrow much from the investigation of actual matter. This is probably largely because actual matter is not (...) continuous, and so it might seem as if discussion of the structure of continuous bodies is merely within the realm of philosophical thought experiments rather than actual scientific investigation. However, classical continuum mechanics models actual matter as if it were continuous, and it has implications about the structure of continuous bodies and about what contact and impenetrability are. This paper describes the relevant notions from classical continuum mechanics so as to resolve the alleged tension between contact and impenetrability. (shrink)

Rotation in a general relativistic framework is examined. This concept, combined with the appropriate mechanical work concept, is used to show how the Euclidean group of transformations, serving as an invariance requirement associated with the principle of objectivity, can lead to erroneous conclusions.

“Special relativity killed the classical dream of using the energy-momentumvelocity relations as a means of probing the dynamical origins of [the mass of the electron]. The relations are purely kinematical” (Pais, 1982, 159). This perceptive comment comes from a section on the pre-relativistic notion of electromagnetic mass in ‘Subtle is the Lord . . . ’, Abraham Pais’ highly acclaimed biography of Albert Einstein. ‘Kinematical’ in this context means ‘independent of the details of the dynamics’. In this paper we examine (...) the classical dream referred to by Pais from the vantage point of relativistic continuum mechanics. (shrink)

Ernst Mayr proposed a distinction between “proximate”, mechanistic, and “ultimate”, evolutionary, causes of biological phenomena. This dichotomy has influenced the thinking of many biologists, but it is increasingly perceived as impeding modern studies of evolutionary processes, including study of “niche construction” in which organisms alter their environments in ways supportive of their evolutionary success. Some still find value for this dichotomy in its separation of answers to “how?” versus “why?”questions about evolution. But “why is A?” questions about evolution necessarily take (...) the form “how does A occur?”, so this separation is illusory. Moreover, the dichotomy distorts our view of evolutionary causality, in that, contra Mayr, the action of natural selection, driven by genotype-phenotype-environment interactions which constitute adaptations, is no less “proximate” than the biological mechanisms which are altered by naturally selected genetic variants. Mayr’s dichotomy thus needs replacement by more realistic, mechanistic views of evolution. From a mechanistic viewpoint, there is a continuum of adaptations from those evolving as responses to unchanging environmental pressures to those evolving as the capacity for niche construction, and intermediate stages of this can be identified. Some biologists postulate an association of “phenotypic plasticity” (phenotype-environment covariation with genotype held constant) with capacity for niche construction. Both “plasticity” and niche construction comprise wide ranges of adaptive mechanisms, often fully heritable and resulting from case-specific evolution. Association of “plasticity” with niche construction is most likely to arise in systems wherein capacity for complex learning and behavioral flexibility have already evolved. (shrink)

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian (1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, (4,5) and Heisenberg (6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck (9–11) ). A satisfactory unitary theory has to involve these constants (...) like the mathematical numbers. Today, Planck's conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Popper's theory (12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables. (shrink)

This paper is part II of a trilogy on the transition from classical particle mechanics to relativistic continuum mechanics that one of the authors is working on. The first part, on the Trouton experiment, was published in the Stachel festschrift (Janssen 2003). This paper focuses on the Lorentz-Poincaré electron, and, in particular, on the "Poincaré pressure" or "Poincaré stresses" introduced to stabilize the electron. It covers both the original argument by Poincaré (1906) and a modern relativistic argument (...) for adding a negative pressure term to the system's energy-momentum tensor inspired by the work of Laue (1911a, b). It highlights the importance of a paper by Lorentz (1899) in this context and of the "electromagnetic mechanics" of Abraham (1903). (shrink)

It is widely agreed that social factors are related to health outcomes: much research served to establish correlations between classes of social factors on the one hand and classes of disease on the other hand. However, why and how social factors are an active part in the aetiology of disease development is something that is gaining attention only recently in the health sciences and in the medical humanities. In this paper, we advance the view that, just as bio-markers help trace (...) the causal continuum from exposure to disease development at the biological level, socio-markers ought to be introduced and studied in order to trace the social continuum from exposure to disease development. We explain how socio-markers differ from social indicators and how they can be used in combination with bio-markers in order to reconstruct the mixed mechanisms of health and disease, namely mechanisms in which both biological and social factors have an active causal role. (shrink)

We present three different derivations of the transition probabilities to the continuum. It is shown that calculations, performed as a direct application of the postulates of orthodox quantum mechanics (OQM), do not yield results consistent with experiments. Traditional treatments are summarized and criticized. The relation of the transitions to the continuum with the traditional quantum measurement problem is pointed out; we sum up and comment some contributions concerning this issue. It is shown that an approach based on (...) the notion of spontaneous projections yields expressions of the transition probabilities similar to those obtained in the traditional way and new predictions which could be submitted to experimental tests. (shrink)

Turing tersely mentioned a notion of ``cultural search'' while otherwise deeply engaged in the design and operations of one of the earliest computers. His idea situated the individual squarely within a collaborative intellectual environment, but did he mean to suggest this in the form of a general information system? In the same writing Turing forecast mechanizations of proofs and outlined genetical searches, much later implemented in cellular automata. The conjecture explores the networked data-information-knowledge continuum as the subject of Turing's (...) notions of search and intelligence, using analogous models from library systems theory. Floridi's philosophy of information is posed as a potential guide to applied information services design of the Turing type. The initial problem is to identify a minimal set of assumptions from Turing's essay beyond the general context of computing. This set will form a bridge to an analogous set of principles in library systems models by eliciting supporting evidence in the literature relating the two. Finally it will be shown how Floridi's philosophy of information more fully encompasses Turing's insight in view of the conjecture. (shrink)

I applaud Fenton and Wiers' attempt to find a demarcation point between cases of addiction that fall within the range of normal function and those that may count as disease. However, I argue that continua don't offer demarcation points, the mechanisms involved are not demonstrably different, and trying to pin down subjectivity doesn't help.

Two prevailing paradigms explain the diversity of sex-determining modes in reptiles. Many researchers, particularly those who study reptiles, consider genetic and environmental sex-determining mechanisms to be fundamentally different, and that one can be demonstrated experimentally to the exclusion of the other. Other researchers, principally those who take a broader taxonomic perspective, argue that no clear boundaries exist between them. Indeed, we argue that genetic and environmental sex determination in reptiles should be seen as a continuum of states represented by (...) species whose sex is determined primarily by genotype, species where genetic and environmental mechanisms coexist and interact in lesser or greater measure to bring about sex phenotypes, and species where sex is determined primarily by environment. To do otherwise limits the scope of investigations into the transition between the two and reduces opportunities to use studies of reptiles to advance understanding of vertebrate sex determination generally. © 2004 Wiley Periodicals, Inc. (shrink)

Since the emergence of computing as a mode of investigation in the sciences, computational approaches have revolutionised many fields of inquiry. Recently in philosophy, the question has begun rendering bit by bit—could computation be considered a deeper fundamental building block to all of reality? This paper proposes a continuum computing construct, predicated on a set of core computational principles: computability, discretisation, stability and optimisation. The construct is applied to the set of most fundamental physical laws, in the form of (...) non-relativistic conservation equations which underpin our governing physics. The discretisation approach divides all of space into a mesh of discrete computational cells, and evolution of data in those cells must obey pre-defined stability conditions from established continuum computing theory, namely, the Courant–Friedrichs–Lewy condition. Evolving cell-state data in a manner which logically optimises computational efficiency, combined with the defined stability condition, the construct derives a central coupling of space, time, and the fastest speed of information propagation. This coupling formed at the lowest level by the computing construct, naturally and inherently produces aspects of special relativity and general relativity at the macroscale. This paper therefore proposes a new explanation of why the nature of space and time may be fused, and explores simple emergent congruities with relativity. The theory invites us to reverse the philosophical premise of the original Simulation Argument. This argument currently leads us to consider: how might all of the observed physics of our reality be reproduced computationally in a simulation of our existence? This work asks instead: could foundations of the physics of our universe—namely spacetime coupling—emerge inherently, and necessarily, from fundamental principles of computation? (shrink)

We formulate a discrete quantum-mechanical precursor to spacetime geometry. The objective is to provide the foundation for a quantum mechanics that is rooted exclusively in quantum-mechanical concepts, with all classical features, including the three-dimensional spatial continuum, emerging dynamically.

Urban dwellers’ food and nutritional wellbeing are both dependent on infrastructure and can be indicative of wider wellbeing in urban contexts and societal health. This paper focuses on the multiple relationships that exist between food and infrastructure to provide a thorough theoretical and empirical grounding to urgent work on urban food security and nutrition in the context of rapid urban and nutrition transitions in the South. We argue that urban systems and food systems thinking have not been well aligned, but (...) that such alignment is not only timely and overdue but also fruitful for both thematic areas of research and policy. We draw in particular on work within wider urban political economy and political ecology that can be classified as part of the ‘infrastructural turn’ that is influential with urban studies but little acknowledged within food studies. Drawing on these literatures helps us to better understand the interrelationships between people, things and ideas that make up both infrastructure and food systems. Policy, planning and research relating to both food and urban systems cannot afford to ignore such interlinkages, though much policy still operates on the neat assumptions of progressive connectivity to ‘the grid’ and formal food retail. Instead we argue how in many urban governance systems, a variety of hybrid mechanisms—on and off the grid, public and private formal and informal—better represent how urban residents, particularly the most marginalised, meet their everyday food and infrastructural needs along a continuum of gridded and off-grid access. (shrink)

There is currently no consensus regarding what measures are most valid to demonstrate perceptual processing without awareness. Likewise, whether conscious perception and unconscious processing rely on independent mechanisms or lie on a continuum remains a matter of debate. Here, we addressed these issues by comparing the time courses of subjective reports, objective discrimination performance and response priming during meta-contrast masking, under similar attentional demands. We found these to be strikingly similar, suggesting that conscious perception and unconscious processing cannot be (...) dissociated by their time course. Our results also demonstrate that unconscious processing, indexed by response priming, occurs, and that objective discrimination performance indexes the same conscious processes as subjective visibility reports. Finally, our results underscore the role of attention by showing that how much attention the stimulus receives relative to the mask, rather than whether processing is measured by conscious discrimination or by priming, determines the time course of meta-contrast masking. (shrink)

The following is an edited version of Roger Penrose's lecture at the Fifth Mind and Brain Symposium at the Institute of Psychiatry, London, on 29 October 1994, introducing the themes of his recent book Shadows of the Mind. The talk begins by outlining some options for the modelling of the relationship between consciousness and computation, and provides evidence for a model in which it is not possible even in principle to simulate mathematical understanding computationally. It is argued that mathematical understanding (...) is on a continuum with consciousness in general, and that non-computability is a feature of all consciousness. The talk then goes on to outline some of the problems of the relationship between quantum and classical physics and proposes a new theory of `objective reduction' by quantum gravity to bridge the explanatory gap. The talk concludes by examining cytoskeletal microtubules as a possible site for quantum-coherent events in the brain. It is suggested that this might be the physical basis of conscious events. (shrink)

The four antinomies of Zeno of Elea continue to be provoking issues that remain relevant for the foundation of science. Aristotle used this antinomy to arrive at a deeper understanding of movement : it is a fluent continuum that he considers to be a whole. The parts, if any, are only potentially present. Similarly, quantum mechanics states that movement is quantized ; things move or change in nonreducible steps, the so-called quanta. This view is in contrast to classical (...) mechanics, where infinitesimally small steps are permitted. The objective of the present study is to show the merits of the Aristotelian approach. Examples from modern science serve to illustrate the philosophical statements. (shrink)

The four antinomies of Zeno of Elea, especially Achilles and the tortoise continue to be provoking issues which are even now not always satisfactory solved. Aristotle himself used this antinomy to develop his understanding of movement: it is a fluent continuum that has to be treated as a whole. The parts, if any, are only potentially present in the whole. And that is exactly what quantum mechanics is claiming: movement is quantized in contrast to classical mechanics. The (...) objective of this study is to show the merits of the Aristotelian approach. It is a good candidate for serving as the philosophical background for understanding fundamental aspects of quantum mechanics. Especially mentioned are the influence of the final state in quantum mechanics that in philosophy could be correlated with the final cause. Like in the work of Aristotle also in this study examples from science are presented to illustrate the philosophical approach. But, in contrast to ancient Greek, the examples now relate to issues which are only fully accessible to the scientifically trained reader. As the main conclusion the dialogue between scientists and philosophers is strongly recommended which will result in progress in both disciplines. (shrink)

Cross-situational learning is a mechanism for learning the meaning of words across multiple exposures, despite exposure-by-exposure uncertainty as to the word's true meaning. We present experimental evidence showing that humans learn words effectively using cross-situational learning, even at high levels of referential uncertainty. Both overall success rates and the time taken to learn words are affected by the degree of referential uncertainty, with greater referential uncertainty leading to less reliable, slower learning. Words are also learned less successfully and more slowly (...) if they are presented interleaved with occurrences of other words, although this effect is relatively weak. We present additional analyses of participants’ trial-by-trial behavior showing that participants make use of various cross-situational learning strategies, depending on the difficulty of the word-learning task. When referential uncertainty is low, participants generally apply a rigorous eliminative approach to cross-situational learning. When referential uncertainty is high, or exposures to different words are interleaved, participants apply a frequentist approximation to this eliminative approach. We further suggest that these two ways of exploiting cross-situational information reside on a continuum of learning strategies, underpinned by a single simple associative learning mechanism. (shrink)

Much of the legal literature discusses regulation and regulatory forms with a seemingly implicit assumption that "those to be influenced" are inherently self-interested and thus motivated to comply with legal structures only when there are sufficient external incentives to do so. This view of the person is inconsistent with recent perspectives in the field of psychology. A law and morality perspective, coupled with insights from the field of psychology, asserts that influence, compliance, and motivation are far more complex than this (...) legal literature would suggest. In this Article, we map the varying influence structures, motives, psychological needs, emotional mechanisms, and levels of moral reasoning that various forms of regulation, from hard law to soft law, might appeal to. We provide examples from global banking and one soft law initiative, the Equator Principles, to illustrate reasons psychology would suggest why soft law may be more effective in some circumstances in influencing behavior within the firm than hard law, while recognizing important limits to such influence. (shrink)

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric (...) of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed. (shrink)

We propose a new point of view for interpreting Newton’s and Einstein’s theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing (...) machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

Thermodynamic macro variables, such as the temperature or volume macro variable, can take on a continuum of allowable values, called thermodynamic macro values. Although referring to the same macro phenomena, the macro variables of Boltzmannian Statistical Mechanics (BSM) differ from thermodynamic macro variables in an important respect: within the framework of BSM the evolution of macro values of systems with finite available phase space is invariably modelled as discontinuous, due to the method of partitioning phase space into macro (...) regions with sharp, fixed boundaries. Conceptually, this is at odds with the continuous evolution of macro values as described by thermodynamics, as well as with the continuous evolution of the micro state assumed in BSM. This discrepancy I call the discontinuity problem. I show how it arises from BSM’s framework and demonstrate its consequences, in particular for the foundational project of reducing thermodynamics to BSM: thermodynamic macro values are shown to not supervene on the corresponding BSM macro values. With supervenience being a conditio sine qua non for the kind of reduction envisaged by the foundational project, the latter is in jeopardy. (shrink)

This thematic issue addresses questions of constraints on the evolution of form—physical, biological, and technical. Here, form is defined as an embodiment of a specific structure, which can be hierarchically different yet emerge from the same processes. The focus of this contribution is about how developmental biology and paleontology can be better integrated and compared in order to produce hypotheses about the evolution of form. The constraints on current EvoDevo research stem from the disconnect in the focus of study for (...) developmental geneticists and evolutionary morphologists; the former being interested in early developmental events at a molecular level in a model animal, the latter in late developmental events or comparison between adult forms, at a structural level in non-model animals. In order to truly integrate information from both fields in our understanding of evolutionary processes, morphology needs to be reintegrated in the study of gene expression, and its time frame needs to be extended beyond early developmental stages. Gene expression in non-model organisms also needs to be studied in order to gain perspective into primitive patterning at evolutionary nodes. Hypotheses formed by the comparison of expression patterns and morphologies seen in extant species can then be tested against forms found in the fossil record, coming closer to understanding the mechanisms underlying evolution. (shrink)

The coronavirus pandemic, like its predecessors - AIDS, Ebola, etc., is evidence of the evolutionary instability of the socio-cultural and ecological niche created by mankind, as the main factor in the evolutionary success of our biological species and the civilization created by it. At least, this applies to the modern global civilization, which is called technogenic or technological, although it exists in several varieties. As we hope to show, the current crisis has less ontological as well as epistemological roots; its (...) reason lies in the main evolutionary trends in the development of science as a social institution. It was only later that epistemological factors were transformed into existential-ontological factors associated with the asymmetry of the existence of civilization and our biosocial nature. The perception or ignorance of a risk factor as a real fact is determined by the presence or absence of knowledge about it. In other words, risk is the result of the integration of the corresponding ontological concept into the general categorical structure. The plurality of such structures is a hallmark of multidisciplinary ontologies, each of which is associated with its own factual continuum. The aim of this study was to conceptually model for support relativistic parameter of evolutionary efficiency of stable evolutionary human strategy (SESH) in its techno-rationalistic module. The meaning of this term is equivalent to the category of scientific and technological development. (shrink)

In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the “helical staircase”, which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan’s discussion, since he (...) argued—but never proved—that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely (a) in 3d Einstein-Cartan gravity—this is Cartan’s case of constant pressure and constant intrinsic torque—and (b) in 3d Poincaré gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory. (shrink)

For quantum mechanics to form the crux of a robust model of divine action, random quantum fluctuations must be amplified into the macroscopic realm. What has not been recognized in the divine action literature to date is the degree to which differential dynamics, continuum mechanics, and condensed matter physics prevent such fluctuations from infecting meso- and macroscopic systems. Once all of the relevant physics is considered, models of divine action based on quantum randomness are shown to be (...) far more limited than is generally assumed. Unless some sort of new physical mechanism is discovered, the amplification problem cannot be solved. (shrink)

Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are (...) classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F. W. Lawvere, who was in fact striving to fashion a decisive axiomatic framework for continuum mechanics.. (shrink)

This article surveys the difficulties in establishing determinism for classical physics within the context of several distinct foundational approaches to the discipline. It explains that such problems commonly emerge due to a deeper problem of ‘missing physics'. The Problems of Formalism Norton's Example Three Species of Classical Mechanics 3.1 Mass point physics 3.2 The physics of perfect constraints 3.3 Continuum mechanics Conclusion CiteULike Connotea Del.icio.us What's this?

This paper examines a fundamental problem in applied mathematics. How can one model the behavior of materials that display radically different, dominant behaviors at different length scales. Although we have good models for material behaviors at small and large scales, it is often hard to relate these scale-based models to one another. Macroscale models represent the integrated effects of very subtle factors that are practically invisible at the smallest, atomic, scales. For this reason it has been notoriously difficult to model (...) realistic materials with a simple bottom-up-from-the-atoms strategy. The widespread failure of that strategy forced physicists interested in overall macro-behavior of materials toward completely top-down modeling strategies familiar from traditional continuum mechanics. The problem of the ``tyranny of scales'' asks whether we can exploit our rather rich knowledge of intermediate micro- scale behaviors in a manner that would allow us to bridge between these two dominant methodologies. Macroscopic scale behaviors often fall into large common classes of behaviors such as the class of isotropic elastic solids, characterized by two phenomenological parameters---so-called elastic coefficients. Can we employ knowledge of lower scale behaviors to understand this universality---to determine the coefficients and to group the systems into classes exhibiting similar behavior? (shrink)

It is contended that one of the promising directions for brooding over the problem of incommensurability of paradigms, coined by T. Kuhn and P. Feyerabend, may be associated with the trend of neo-Kantian epistemology, embodied by the writings of Ernst Cassirer. According to Cassirer, the statements fixing connections and relationships between mathematical ideal constructs render a reliable ‘neutral language’ that can serve as a firm ground for comparing the ‘old’ and ‘new’ paradigms. The aim of the paper is to offer (...) Maxwell’s work as a test case against the incommensurability thesis. A case study of the genesis and functioning of a neutral mathematical language related to the Maxwellian solid synthesis of optics and electromagnetism is provided. It is elicited that its basis is constituted by stodgy language of continuum mechanics. The main function of the neutral language was to project the consequences of all the unified partial theoretical schemes (‘old’ rival paradigms) of Ampѐre, Faraday, Biot, Savar et al. onto the grand mathematical model, ‘rewrite’ all known laws in this pervasive mathematical language, compare their conclusions with each other to eliminate contradictions. Eventually the stuff was generalized in a self-consistent system of Maxwell’s equations. (shrink)

The Gauss-Hertz principle is extended by the use of existence conditions (or constraints) to obtain a hierarchy of differential equations which include all classical equations of continuum mechanics (including electrodynamics) and the harmonic oscillator potential as well.

Earthquake and Volcano Deformation is the first textbook to present the mechanical models of earthquake and volcanic processes, emphasizing earth-surface deformations that can be compared with observations from Global Positioning System receivers, Interferometric Radar, and borehole strain- and tiltmeters. Paul Segall provides the physical and mathematical fundamentals for the models used to interpret deformation measurements near active faults and volcanic centers. Segall highlights analytical methods of continuum mechanics applied to problems of active crustal deformation. Topics include elastic dislocation (...) theory in homogeneous and layered half-spaces, crack models of faults and planar intrusions, elastic fields due to pressurized spherical and ellipsoidal magma chambers, time-dependent deformation resulting from faulting in an elastic layer overlying a viscoelastic half-space and related earthquake cycle models, poroelastic effects due to faulting and magma chamber inflation in a fluid-saturated crust, and the effects of gravity on deformation. He also explains changes in the gravitational field due to faulting and magmatic intrusion, effects of irregular surface topography and earth curvature, and modern concepts in rate- and state-dependent fault friction. This textbook presents sample calculations and compares model predictions against field data from seismic and volcanic settings from around the world. Earthquake and Volcano Deformation requires working knowledge of stress and strain, and advanced calculus. It is appropriate for advanced undergraduates and graduate students in geophysics, geology, and engineering. Professors: A supplementary Instructor's Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: https://press.princeton.edu/class_use/solutions.html. (shrink)

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles have not come (...) from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. The simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in: (1) clarifying and guiding further research in the mathematical sciences, but also in (2) reforming pedagogy and popularization in ways that will not prove deceptive. (shrink)

The present book of proceedings includes chapters related to the areas of pure, applied and inter-disciplinary mathematics reflecting the potential applications in the domains of sciences and engineering. The main areas include algebra and its applications, analysis and approximation theory, cryptography, computational fluid dynamics, continuum mechanics and vibrations, differential equations and applications, graph theory, fuzzy mathematics and logic, numerical analysis, optimization and its applications, wave propagation, etc. The scientists, engineers, academicians and researchers working in the proposed areas of (...) coding and information theory, computational fluid dynamics, differential equations, fuzzy sets and systems, numerical analysis, optimization, vibrations, etc. looking for new insight and ideas in these areas will be benefitted by the contents of this book. In fact, it will be useful to a large class of readers interested in recent findings related to mathematical sciences and their applications to the diverse domains of knowledge. The book will be a fruitful contribution to all knowledge seekers and researchers in the concerned areas who are looking for new insight and ideas. As it consists of updated research articles on emerging areas of mathematical sciences and their applications, it will be a valued addition to the learning resource centres of various universities, industries, and research and development organizations across the globe. (shrink)

This book provides instances of what the technology and semantic field of music have contributed to the development of epistemology, logic and the early modern sciences of developmental biology, continuum mechanics anatomy and physiological psychology, as well as what some other domains have given back to the philosophy and theory of music.

In his recent book, Explaining Chaos, Peter Smith presents a new problem in the foundations of chaos theory. Specifically, he argues that the standard ways of justifying idealizations in mathematical models fail when it comes to the infinite intricacy found in strange attractors. I argue that Smith's analysis undermines much of the explanatory power of chaos theory. A better approach is developed by drawing analogies from the models found in continuum mechanics.

Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The (...) Poincaré gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article. (shrink)

The continuum form of the Gauss-Hertz principle is extended to include the time domain as well as space. The Schrödinger equation and general relativity are derived by this method. The equivalence of the principle is shown to that of the Hamiltonian method where the energy is the expression −[φ∇ 2 φ+A·∇2 A], with φ being the difference between the acceleration potential and potential energy density, andA being the difference between the vector potentials of the acceleration field and the force (...) field. The goal of Hertz to “demonstrate a third arrangement of the principles of mechanics...which starts with... time, space and mass” has apparently been achieved for relativity and for quantum mechanics, in addition to those classical equations previously found. (shrink)

Parallel lives is an ontological model of nature in which quantum mechanics and special relativity are unified in a single universe with a single space-time. Point-like objects called lives are the only fundamental objects in this space-time, and they propagate at or below c, and interact with one another only locally at point-like events in space-time, very much like classical point particles. Lives are not alive in any sense, nor do they possess consciousness or any agency to make decisions—they (...) are simply point objects which encode memory at events in space-time. The only causes and effects in the universe occur when lives meet locally, and thus the causal structure of interaction events in space-time is Lorentz invariant. Each life traces a continuous world-line through space-time, and experiences its own relative world, fully defined by the outcomes of past events along its world-line, which are encoded in its external memory. A quantum field comprises a continuum of lives throughout space-time, and familiar physical systems like particles each comprise a sub-continuum of the lives of the field. Each life carries a hidden internal memory containing a local relative wavefunction, which is a local piece of a pure universal wavefunction, but it is the relative wavefunctions in the local memories throughout space-time which are physically real in PL, and not the universal wavefunction in configuration space. Furthermore, while the universal wavefunction tracks the average behavior of the lives of a system, it fails to track their individual dynamics and trajectories. There is always a preferred separable basis, and for an irreducible physical system, each orthogonal term in this basis is a different relative world—each containing some fraction of the lives of the system. The relative wavefunctions in the lives’ internal memories govern which lives of different systems can meet during future local interactions, and thereby enforce entanglement correlations—including Bell inequality violations. These, and many other details, are explored here, but several aspects of this framework are not yet fleshed out, and work is ongoing. (shrink)

The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, (...) closer to what is actually done in experimental measurements and computations. The method replaces C by C n which is the set of pairs, R n,I n, of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on R n is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also R n based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on C n and on R n space are briefly investigated. (shrink)

To investigate mechanisms of rational representation, I consider construction of an ordered continuum of psychophysical scale of magnitude of sensation; counting mechanism leading to an approximate numerosity scale for integers; and conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

This paper shows an analysis of some found solutions for the famous aporia of the race between Achilles and the Tortoise. As an introduction, we present the mechanical solution, to establish that it is not in the field of matters of fact where you can resolve a purely rational problem like the one raised by Zeno of Elea. And so, the main part of the article is dedicated to the mathematical solutions, which face the problem under the point of view (...) of the single, mathematical reason. There are two mathematical solutions for Zeno’s paradox. First, we attend to that which we denominate “classical” (because it is the most habitually used by mathematicians), which is based on the calculus of addiction of terms of a geometrical series in progressive decrease. The second considered solution is the one that was proposed by Russell, based upon the theory of the transfinite numbers. The analysis we have made gets us to discover that no one of those solutions can save itself from falling into logical contradictions, therefore it seems that Zeno’s problem is an authentic aporia, which, after so much time, continues challenging human intelligence. As the article’s conclusion, we suggest that the cause of the impossibility of solving Zeno’s problem is the very notion of mathematical continuum, because this notion infringes the logic of facts. (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)

Orthodox quantum mechanics is technically built around an element that von Neumann called Process 1. In its basic form it consists of an action that reduces the prior state of a physical system to a sum of two parts, which can be regarded as the parts corresponding to the answers ‘Yes’ and ‘No’ to a specific question that this action poses, or ‘puts to nature’. Nature returns one answer or the other, in accordance with statistical weightings specified by the (...) theory. Thus the standard statistical element in quantum theory enters only after the Process-1 choice is made, while the known deterministic element in quantum theory governs the dynamics that prevails between the reduction events, but not the process that determines which of the continuum of allowed Process-1 probing actions will actually occur. The rules governing that selection process are not fixed by the theory in its present form. This freedom can be used to resolve in a natural way an apparent problem of the orthodox theory, its biocentrism. That resolution produces a rationally coherent realization of the theory that preserves the basic orthodox structure but allows naturally.. (shrink)