This book examines the interface between mathematics and applied sciences. The editor examines the present and future needs for the interaction between various science and engineering areas. This edited book brings together the cutting-edge research on mathematics, combining various fields of science and engineering. The book begins with an introduction to computing and modeling. Next, computation and modeling trends are covered, along with chapters on structural static and vibration problems, heat conduction and diffusion problems, and fluid dynamics problems. Soft computing (...) and machine intelligence based modeling are also discussed, along with bio medical problems. The book concludes with a chapter on modeling in mining, electrical, electronics, and chemical problems. This book is a reference for students and researchers in various fields of engineering such as computer, mechanical, civil, aerospace, electrical, and chemical, as well as and other sciences such as applied mathematics, physics, chemistry, and life sciences. (shrink)

According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the concrete world (...) is just as it in fact is, then T’ bear on this claim. It concludes that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former sort as explanatorily bad, this reason does not apply to theories of the latter sort. (shrink)

The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is (...) considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization. (shrink)

This textbook offers the first unified treatment of wave propagation in electronic and electromagnetic systems and introduces readers to the essentials of the transfer matrix method, a powerful analytical tool that can be used to model and study an array of problems pertaining to wave propagation in electrons and photons. It is aimed at graduate and advanced undergraduate students in physics, materials science, electrical and computer engineering, and mathematics, and is ideal for researchers in photonic crystals, negative index materials, left-handed (...) materials, plasmonics, nonlinear effects, and optics. Peter Markos and Costas Soukoulis begin by establishing the analogy between wave propagation in electronic systems and electromagnetic media and then show how the transfer matrix can be easily applied to any type of wave propagation, such as electromagnetic, acoustic, and elastic waves. The transfer matrix approach of the tight-binding model allows readers to understand its implementation quickly and all the concepts of solid-state physics are clearly introduced. Markos and Soukoulis then build the discussion of such topics as random systems and localized and delocalized modes around the transfer matrix, bringing remarkable clarity to the subject. Total internal reflection, Brewster angles, evanescent waves, surface waves, and resonant tunneling in left-handed materials are introduced and treated in detail, as are important new developments like photonic crystals, negative index materials, and surface plasmons. Problem sets aid students working through the subject for the first time. (shrink)

In this paper, we construct a parallel image of the conventional Maxwell theory by replacing the observer-time by the proper-time of the source. This formulation is mathematically, but not physically, equivalent to the conventional form. The change induces a new symmetry group which is distinct from, but closely related to the Lorentz group, and fixes the clock of the source for all observers. The new wave equation contains an additional term (dissipative), which arises instantaneously with acceleration. This shows that the (...) origin of radiation reaction is not the action of a “charge” on itself but arises from inertial resistance to changes in motion. This dissipative term is equivalent to an effective mass so that classical radiation has both a massless and a massive part. Hence, at the local level the theory is one of particles and fields but there is no self-energy divergence (nor any of the other problems). We also show that, for any closed system of particles, there is a global inertial frame and unique (invariant) global proper-time (for each observer) from which to observe the system. This global clock is intrinsically related to the proper clocks of the individual particles and provides a unique definition of simultaneity for all events associated with the system. We suggest that this clock is the historical clock of Horwitz, Piron, and Fanchi. At this level, the theory is of the action-at-a-distance type and the absorption hypothesis of Wheeler and Feynman follows from global conservation of energy. (shrink)

The coordinated attack scenario and the electronic mail game are two paradoxes of common knowledge. In simple mathematical models of these scenarios, the agents represented by the models can coordinate only if they have common knowledge that they will. As a result, the models predict that the agents will not coordinate in situations where it would be rational to coordinate. I argue that we should resolve this conflict between the models and facts about what it would be rational to (...) do by rejecting common knowledge assumptions implicit in the models. I focus on the assumption that the agents have common knowledge that they are rational, and provide models to show that denying this assumption suffices for a resolution of the paradoxes. I describe how my resolution of the paradoxes fits into a general story about the relationship between rationality in situations involving a single agent and rationality in situations involving many agents. (shrink)

The Upanishads reveal that in the beginning, nothing existed: “This was but non-existence in the beginning. That became existence. That became ready to be manifest”. (Chandogya Upanishad 3.15.1) The creation began from this state of non-existence or nonduality, a state comparable to (0). One can add any number of zeros to (0), but there will be nothing except a big (0) because (0) is a neutral number. If we take (0) as Nirguna Brahman (God without any form and attributes), then (...) from where and how did the universe come into existence? -/- The neutral power (0) cannot produce anything without having an element of duality in it. Although Nirguna Brahman is neutral, it has a positive, negative, and neutral pole, constituting its Prakriti or nature. Prakriti has three latent Gunas (modes or qualities): Satva, Rajas, and Tamas. They are related to Gyana Shakti (the power of knowledge) Sankalpa Shakti (the power of ideation), and Kriya Shakti (the power of action). Science says that Atom is the basic element from which the universe evolved. The Atom has three nuclei- electron, pluton, and neutron. The Satva, Rajas, and Tamas in Indian spirituality are nothing but the mystical names for the nuclei of an atom. -/- According to Bhagavata Purana, Prakriti is also constituted of the elements of Time, Karma (action/destiny) and Swabhava (innate nature). Time disturbs the equilibrium of Gunas. From the aspect of Karma is produced an entity called Mahat, in the form of intelligence. Mahat is dominated by Satva and Rajas. From Mahat manifests the next evolute dominated by Tamas with three predominant qualities – Dravya (substance), Kriya (action), as well as intelligence. It forms to become the Ego principle (Ahamkara). -/- Ahamkara has three modes – Satvic, Rajasic and Tamasic. Satvic is Jnana oriented, Rajasic action-oriented and Tamasic Dravya (substance) oriented. From the Tamasic Ahankara, five gross elements are produced (ether, air, fire, water, and earth - Akash, Vayu, Agni, Jal and Prithvi). From the Satvic Ahamkara, guardian deities (the sun and moon, deities of sense organs, and organs of action) and from Rajasic Ahamkara, ten senses (Indriyas), five senses of perception, five organs of action, the faculties of intellect and Prana (life breath) are produced. -/- Bhagavata says that since these energies, elements, and faculties remained disassociated, they were combined to form a Cosmic Egg. The egg floats in the primal waters for a thousand years. Then God enters this Cosmic Egg and manifests himself as the Cosmic Purusha. He is the first nucleus, the God particle equivalent to the number (1) which is the embodiment of everything in the universe. The concept of creation and dissolution in Hinduism can be compared to the waves in an ocean that appear and disappear incessantly. The Manvantaras are such successive episodes of creation emerging from the Cosmic Person, the Manu who is embodied God-Consciousness, God himself. These episodes of creation are measured in Hinduism in terms of Manvantaras, the epochs of Manus. -/- Manu is the ‘First-Born’ (1) of God, the Cosmic Purusha from whom the world has originated. This Cosmic Person (Purusha) has been described as having fourteen biospheres (heavens) that are inhabited by various life forms in the order of evolution of consciousness. If we begin from the human biosphere (Bhuloka), there are ten astral biospheres that a man must transcend to attain liberation or Mukti from the cycle of births and deaths. -/- The number (1) produces the primary numbers up to (9) through the transmutation of the creational energies and qualities. The process can be related to the descent and cyclical evolution of (1) through ten spiritual stages, finally merging with the nondual (0). The number (10) marks the merger of (1) with (0). The Hindu worldview is that all life forms in an episode of creation must evolve to become one with the non-dual state (0) to attain liberation from the cycle of births and deaths in a cyclical process. -/- . (shrink)

Dirac’s relativistic theory of electron generally results in two possible solutions, one with positive energy and other with negative energy. Although positive energy solutions accurately represented particles such as electrons, interpretation of negative energy solution became very much controversial in the last century. By assuming the vacuum to be completely filled with a sea of negative energy electrons, Dirac tried to avoid natural transition of electron from positive to negative energy state using Pauli’s exclusion principle. However, many scientists (...) like Bohr objected to the idea of sea of electrons as it indicates infinite density of charge and electric field and consequently infinite energy. In addition, till date, there is no experimental evidence of a particle whose total energy (kinetic plus rest) is negative. In an alternative approach, Feynman, in quantum field theory, proposed that particles with negative energy are actually positive energy particles running backwards in time. This was mathematically consistent since quantum mechanical energy operator contains time in denominator and the negative sign of energy can be absorbed in it. However, concept of negative time is logically inconsistent since in this case, effect happens before the cause. To avoid above contradictions, in this paper, we try to reformulate the Dirac’s theory of electron so that neither energy needs to be negative nor the time is required to be negative. Still, in this new formulation, two different possible solutions exist for particles and antiparticles (electrons and positrons). (shrink)

The theory suggests that quantum computations in brain neuronal dendritic-somatic microtubules regulate axonal firings to control conscious behavior. Within microtubule subunit proteins, collective dipoles in arrays of contiguous amino acid electron clouds enable suitable for topological dipole able to physically represent cognitive values, for example, those portrayed by Pothos & Busemeyer (P&B) as projections in abstract Hilbert space.

From electronic design problems to resolution proofs to SAT solvers, this book focuses on the satisfiabilityof theories that consist of propositional logic ...

A macroscopic realization of the peculiar virtual particles is presented. The classical Helmholtz and the Schrödinger equations are differential equations of the same mathematical structure. The solutions with an imaginary wave number are called evanescent modes in the case of elastic and electromagnetic fields. In the case of non-relativistic quantum mechanical fields they are called tunneling solutions. The imaginary wave numbers point to strange consequences: The waves are non-local, they are not observable, and they are described as virtual particles. (...) During the last two decades QED calculations of the solutions with an imaginary wave number have been experimentally confirmed for phonons, photons, and electrons. The experimental proofs of the predictions of non-relativistic quantum mechanics and the Wigner phase time approach for the elastic, electromagnetic and Schrödinger fields will be presented in this article. The results are zero time in the barrier and an interaction time (i.e. a phase shift) at the barrier interfaces. The measured tunneling time scales approximately inversely with the particle energy. Actually, the tunneling time is given only by the barrier boundary interaction time, as zero time is spent inside a barrier. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are (...) mathematical objects. (shrink)

An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including (...) new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law. (shrink)

I argue that the role played by infons in the kind of mathematical theory of information being developed by several workers affiliated to CSLI is analogous to that of the various number systems in mathematics. In particular, I present a mathematical construction of infons in terms of representations and informational equivalences between them. The main theme of the paper arose from an electronic mail exchange with Pat Hayes of Xeroxparc. The exposition derives from a talk I gave at (...) theTheories of Partial Information conference held at the University of Texas at Austin, January 1990. (shrink)

In this work, a neo-classical relativistic mechanics theory is presented where the spin of an electron is an inherent part of its world space-time path as a point particle. The fourth-order equation of motion corresponds to the same covariant Lagrangian function in proper time as in special relativity except for an additional spin energy term. The theory provides a hidden-variable model of the electron where the dynamic variables give a complete description of its motion, giving a classical mechanics (...) explanation of the electron’s spin, its dipole moments, and Schrödinger’s zitterbewegung, These features are also described mathematically by quantum mechanics theory, of course, but without any physical picture of an underlying reality. The total motion of the electron can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called here the spin center. The global motion is sub-luminal and described by Newton’s Second Law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac’s velocity operators having magnitude c. The local spin motion is an inherent perpetual motion, which for a free electron is periodic at the ultra-high zitterbewegung frequency and its path is circular in a spin-center reference frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron’s point charge. The electric dipole energy corresponds to the spin-orbit coupling term involving the electric field that appears in the corrected Pauli non-relativistic Hamiltonian, which has long been used to explain the doublet structure of the spectral lines of the excited hydrogen atom. Pauli’s spin-orbit term is usually derived, however, from his magnetic dipole energy term, including also the effect of Thomas precession, which halves this energy. The magnetic dipole energy from Pauli’s and Dirac’s theory is twice that in the neo-classical theory, a discrepancy that has not been resolved. By defining a spin tensor as the angular momentum of the electron’s total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut–Zanghi electron theory. This allows the equations of motion to be expressed in an equivalent form involving operators applied to a state function of proper time satisfying a neo-classical Dirac–Schrödinger spinor equation. This state function produces the dynamic variables from the same operators as in Dirac’s theory for the electron but without any probability implications. It leads to a neo-classical wave function that satisfies Dirac’s relativistic wave equation for the free electron by applying the Lorentz transformation to express proper time in the state function in terms of an observer’s space-time coordinates, showing that there is a close connection between the neo-classical theory and quantum mechanics theory for the electron’s dynamics. (shrink)

Taxonomy Based modeling was applied to describe drivers’ mental models of variable message signs (VMS’s) displayed on expressways. Progress in road telematics has made it possible to introduce variable message signs (VMS’s). Sensors embedded in the carriageway every 500m record certain variables (speed, flow rate, etc.) that are transformed in real time into “driving times” to a given destination if road conditions do not change.VMS systems are auto-regulative Man-Machine (AMMI) systems which incorporate a model of the user: if the traffic (...) flow is too high, then drivers should choose alternative routes. In so doing, the traffic flow should decrease. The model of the user is based on suppositions such as: people do not like to waste time, they fully understand the displayed messages, they trust the displayed values, they know of alternative routes. However, people also have a model of the way the system functions. And if they do not believe the contents of the message, they will not act as expected.We collected data through interviews with drivers using the critical incidents technique (Flanagan, 1985). Results show that the mental models that drivers have of the way the VMS system works are various but not numerous and that most of them differ from the“ideal expert” mental model. It is clear that users don’t have an adequate model of how the VMS system works and that VMS planners have a model of user behaviour that does not correspond to the behaviour of the drivers we interviewed. Finally, Taxonomy Based Modeling is discussed as a tool for mental model remediation. (shrink)

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to (...) be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions. (shrink)

The double slit experiment for a massive scalar particle is described using intuitionistic logic with quantum real numbers as the numerical values of the particle's position and momentum. The model assigns physical reality to single quantum particles. Its truth values are given open subsets of state space interpreted as the ontological conditions of a particle. Each condition determines quantum real number values for all the particle's attributes. Questions, unanswerable in the standard theories, concerning the behaviour of single particles in the (...) experiment are answered. (shrink)

In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means finished. We present and discuss our proposal in order to get feedback from the Calculemus community.

George Spencer Brown, a polymath and author of Laws of Form, brought together mathematics, electronics, engineering and philosophy to form an unlikely bond. This book investigates Design with the NOR, the title of the yet unpublished 1961 typescript by Spencer Brown. The typescript formed through the author's experiences as technical engineer and developer of a new form of switching algebra for Mullard Equipment LTD., a British manufacturer of electronic components. Related essays contextualise the typescript drawing on a variety of backgrounds (...) from mathematics and engineering to philosophy and sociology. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

Theory of Computation -- Mathematical Logic and Formal Languages.

This book explains, in lay terms, the surprisingly simple system of mathematical logic used in digital computer circuitry. Anecdotal in its style and often funny, it follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. ONES AND ZEROS will be enjoyed by anyone who has a general interest in science and technology.

The development of machine learning and deep learning (DL) in the field of AI (artificial intelligence) is the direct result of the advancement of nano-electronics. Machine learning is a function that provides the system with the capacity to learn from data without being programmed explicitly. It is basically a mathematical and probabilistic model. DL is part of machine learning methods based on artificial neural networks, simply called neural networks (NNs), as they are inspired by the biological NNs that constitute (...) organic brains. Despite its similarity to biological organs such as human brains, major problems arise in trying to attribute moral responsibility to autonomic systems based on hardware including nano-electronic devices, which are sought to replace humans (moral agents) in the context of AI. It is suggested that the required emotional environment which enables actions according to reasons in humans is not witnessed at AI devices. Though AI technology raises an enticing resemblance to human actions, this resemblance is to be considered with a skeptical eye. It is because actions are associated with reasons while causes are connected to operations. Human agents are capable of acting upon their reasons, while AI devices are limited only to operations as they are conditioned by their programming, which is considered as an embodiment of some causes. As moral responsibility goes hand in hand with the capacity to act (according to reasons), attributing moral responsibility to AI devices is revealed to be but a misleading metaphor. (shrink)

Historical accounts of the work of J. J. Thomson find a contradiction in his work. On the one hand, he is presented as a Maxwellian theoretical physicist dealing with a typically Victorian entity, the ether. On the other hand, the analysis of his experimental work at the Cavendish seems to have little connection with his mathematical work. In this paper, I discuss the metaphysical views of J. J. Thomson, and argue that his deep belief in the ultimate continuity of (...) matter can be seen to give a framework to both his theoretical and his experimental work. His metaphysical beliefs were not in the least shaken by the discovery of discrete phenomena and entities, not even by his suggestion of the existence of corpuscles later known as electrons. His formation in Cambridge, together with some ideas that he acquired in his youth at Owens College, Manchester, are the key to understanding his metaphysics and the role it plays in his scientific work. (shrink)

The mathematical framework of Relativistic Schrödinger Theory (RST) is generalized in order to include the self-interactions of the particles as an integral part of the theory (i.e. in a non-perturbative way). The extended theory admits a Lagrangean formulation where the Noether theorems confirm the existence of the conservation laws for charge and energy–momentum which were originally deduced directly from the dynamical equations. The generalized RST dynamics is applied to the case of some heavy helium-like ions, ranging from germanium (Z=32) (...) to bismuth (Z=83), in order to compute the interaction energy of the two electrons in their ground-state. The present inclusion of the electron self-energies into RST yields a better agreement of the theoretical predictions with the experimental data. (shrink)

The mathematical genius Alan Turing, well known for his crucial wartime role in breaking the ENIGMA code, was the first to conceive of the fundamental principle of the modern computer. This text contains first hand accounts by Turing and by the pioneers of computing who worked with him on his revolutionary design for an electronic computing machine - his Automatic Computing Engine.

This monograph presents the latest findings from a long-term research project intended to identify the physics behind Quantum Mechanics. A fundamental theory for quantum mechanics is constructed from first physical principles, revealing quantization as an emergent phenomenon arising from a deeper stochastic process. As such, it offers the vibrant community working on the foundations of quantum mechanics an alternative contribution open to discussion. The book starts with a critical summary of the main conceptual problems that still beset quantum mechanics. The (...) basic consideration is then introduced that any material system is an open system in permanent contact with the random zero-point radiation field, with which it may reach a state of equilibrium. Working from this basis, a comprehensive and self-consistent theoretical framework is then developed. The pillars of the quantum-mechanical formalism are derived, as well as the radiative corrections of nonrelativistic QED, while revealing the underlying physical mechanisms. The genesis of some of the central features of quantum theory is elucidated, such as atomic stability, the spin of the electron, quantum fluctuations, quantum nonlocality, and entanglement. The theory developed here reaffirms fundamental scientific principles such as realism, causality, locality and objectivity. (shrink)

The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s contributions and (...) its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large. (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his (...) class='Hi'>electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his (...) class='Hi'>electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

It is widely assumed that there exist certain objects which can in no way be distinguished from each other, unless by their location in space or other reference-system. Some of these are, in a broad sense, 'empirical objects', such as electrons. Their case would seem to be similar to that of certain mathematical 'objects', such as the minimum set of manifolds defining the dimensionality of an R -space. It is therefore at first sight surprising that there exists no branch (...) of mathematics, in which a third parity-relation, besides equality and inequality, is admitted; for this would seem to furnish an appropriate model for application to such instances as these. I hope, in this work, to show that such a mathematics in feasible, and could have useful applications if only in a limited field. The concept of what I here call 'indistinguishability' is not unknown in logic, albeit much neglected. It is mentioned, for example, by F. P. Ramsey [1] who criticizes Whitehead and Russell [2] for defining 'identity' in such a way as to make indistinguishables identical. But, so far as I can discover, no one has made any systematic attempt to open up the territory which lies behind these ideas. What we find, on doing so, is a body of mathematics, offering only a limited prospect of practical usefulness, but which on the theoretical side presents a strong challenge to conventional ideas. (shrink)