I characterize Bishop's constructive mathematics as an alternative to classical mathematics, which makes use of the actual infinity. From the history an accurate investigation of past physical theories I obtianed some ones - mainly Lazare Carnot's mechanics and Sadi Carnot's thermodynamics - which are alternative to the dominant theories - e.g. Newtopn's mechanics. The way to link together mathematics to theoretical physics is generalized and some general considerations, in particualr on the geoemtry in theoretical physics, are (...) obtained.that. (shrink)

I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. 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.. (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 book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

Context: The dominant approach to the study of perception is representational/computational, with an emphasis on the achievements of the brain and the nervous system, which are taken to construct internal models of the world. Alternatives include ecological, embedded, embodied, and enactivist approaches, all of which emphasize the centrality of action in understanding perception. Problem: Despite sharing many theoretical commitments that lead to a rejection of the classical approach, the alternatives are characterized by important contrasts and points of divergence. Here (...) we focus on the enactive and ecological approaches, in particular, on how they construe the status of the environment and the content of perception. Method: We begin with a review of James Gibson’s ecological psychology, highlighting it as a psychology for all organisms not just humans. Against this backdrop, we consider enactivist arguments against direct perception - a central assertion of the ecological approach - and in favor of interpreting the activity of perceptual agents as a kind of construction of a perceptually meaningful world. Results: We assess the merits of this interpretation and we conclude that it cannot be grounded on fundamental principles such as thermodynamics and organism-environment mutuality. Implications: As a consequence, enactivism remains close to representationalism and entails a form of dualism. Constructivist content: We advance a criticism of the constructivist foundations of the enactive approach to perception. Perception-action mutuality at the heart of enactivism does not require mental construction; indeed, perception-action mutuality obviates construction. (shrink)

Classic examples of ostensive geometrical constructions are used to clarify Kant’s account of how they provide knowledge of claims about rigid bodies we can observe and manipulate. It is argued that on Kant’s account claims warranted by ostensive constructions must be limited to scales and tolerances corresponding to our perceptual competencies. This limitation opens the way to view Riemann’s work as contributing valuable conceptual resources for extending geometrical knowledge beyond the bounds of observation. It is argued that neither Reichenbach’s descriptions (...) of non-Euclidean visualization nor his arguments for conventionalism about geometry undercut this view of Kant’s account of geometrical knowledge. (shrink)

Throughout his life, Hegel showed great interest in physics and mathematics. His most sustained, surviving treatment of Euclidean geometry is his early work ‘Geometrische Studien’, which he completed while he was a private tutor [Hoffmeister] in Frankfurt, shortly before leaving for Jena to join Schelling.GSis not easy reading, but despite that, it seems to me that Hegel presents in it a remarkably erudite as well as interesting and insightful critique of geometry. He investigates some of the themes from (...) the foundations of geometry, in particular from the first book of theElements of Euclid. Like the mathematical philosophies of Kant and Frege, Hegel's understanding of geometry is conceptually based, but unlike them, it is also grounded in the classical Greek philosophy of mathematics, which achieved its definitive expression in Proclus's great commentary onEuclid 1. Much of this classical philosophy of geometry is forgotten nowadays, under the influence of the great modern mathematical philosophers. In my view, it well deserves reconsideration, especially since, as illustrated by Gödel's incompleteness theorems, modern mathematical philosophy has failed in its attempt to ground mathematics within the framework of formal systems.Much ofGShas not survived, and what remains is condensed and fragmentary. It seems that originally, Hegel covered all of the propositions ofEuclid 1rather than just the 14 propositions that are covered in what remains of the originalGS. I have given detailed treatment ofGStogether with related material in Hegel's Jena dissertation elsewhere. The objective of the present paper is to introduce the translation ofGS. (shrink)

The foci of this penetrating study are Euclid's geometry and Descartes' mathematics. It is a contribution to the history of mathematics, but it is much more, for the differing approaches to mathematics in the ancient and the modern worlds is shown to have deep consequences for both doing and knowing. The investigation is centered on the nature of geometrical construction in ancient and modern mathematics, and, by extension, the crucial importance of construction to the reality and self-understanding of modernity.

In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to (...) the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above. (shrink)

In this work a mechanistic explanation of the classical algae growth model built by M. R. Droop in the late sixties is proposed. We first recall the history of the construction of the “predictive” variable yield Droop model as well as the meaning of the introduced cell quota. We then introduce some theoretical hypotheses on the biological phenomena involved in nutrient storage by the algae that lead us to a “conceptual” model. Though more complex than Droop’s one, our model (...) remains accessible to a complete mathematical study: its confrontation to the Droop model shows both have the same asymptotic behavior. However, while Droop’s cell quota comes from experimental bio-chemical measurements not related to intra-cellular biological phenomena, its analogous in our model directly follows our theoretical hypotheses. This new model should then be looked at as a re-interpretation of Droop’s work from a theoretical biologist’s point of view. (shrink)

Recall, if you will, the standard objections to the traditional doctrines. While the most subtle of the competing doctrines is, in my opinion, the Aristotelian and scholastic account of abstraction, the objection to this doctrine is that it requires a realism which is too immediate, so that the forms of one's present state of knowledge are allowed to pass as the forms of nature. And although, as I understand it, Aristotelian mathematics is gained by abstraction from an already fairly abstract (...) matter, one naturally expects in this context one true geometry, rather than alternate geometries. At the same time the strength of this view lies in its confidence that our abstractions must refer to the real world. The defect of the standard Lockian doctrine is that it contains an inner inconsistency, the nature of which we shall notice in due course. And upon application the doctrine explodes at once. On the other hand, the strength of the Lockian doctrine, as well as the doctrine of Berkeley and Hume, lies in its reluctance to allow needless entities entrance into reality. The weakness of the doctrine of Berkeley and Hume is that in moving from the impossible image of Locke's doctrine, the emphasis on particularity requires the agency of custom or habit to join the particulars of sense into classes. But the edges of such classes remain always indefinite. Habit, or custom, would seem not to allow the clean separation of abstractions from each other which the results of intellection give us reason to demand in any adequate doctrine. The weakness of the Kantian doctrine lies in its inability to explicate an antecedent reality; that is, in affirming at last the incognizable thing in itself; while the strength of this doctrine derives from its ability to allow the construction of precise abstractions in a somewhat autonomous manner. (shrink)

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin (...) of the Schrödinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained. (shrink)

For certain methodological and historical reasons, the science of probability (probabilistics) had never been constructed before as a single whole, and it has basically split into probability theory and into statistics. One of the reasons was the neglect of an extremely important methodological principle which reads: It is necessary to distinguish strictly between concrete objects and abstract objects. This principle is displayed and exemplified. Its use has made it possible to discover the basic phenomenon of probalilistics and to construct the (...) science, whose outlines are given. The application of probabilistics to physics gives rise to probabilistic physics, whose particular domains are, among others, both classical statistical physics and quantum physics. The actual meaning of quantum physics becomes quite clear and no artificial interpretation of it proves to be necessary. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Poetics of PerspectiveHarvey L. HixThe Poetics of Perspective, by James Elkins; xv & 324 pp. Ithaca: Cornell University Press, 1994, $39.95.The Poetics of Perspective does not mention that Leonardo was born more than 100 years before Galileo and nearly 200 before Newton, but doing so would underscore its thesis. According to James Elkins, our anachronistic view of perspective, invented in the Enlightenment, systematically distorts our understanding of (...) Renaissance art. Instead of “tracing the outline” of the real nature of Renaissance art, our discussions (to borrow Wittgenstein’s metaphor) merely trace “the frame through which we look at it.” The actual Renaissance view, unlike the modern idée fixe, involved no concept of a unified pictorial space. In the absence of such a concept, “Renaissance authors and artists [End Page 368] thought there were many compatible perspectives, so that their writing and painting evince a ‘pluralist’ approach in strict contrast to the monolithic mathematical perspective we imagine today” (p. xi). The Poetics of Perspective attempts, in six substantial chapters and a brief envoi, to unearth this pluralistic Renaissance approach to perspective buried beneath modern preconceptions.Elkins identifies a number of differences between the Renaissance view of perspective and modern views. For instance, no longer is perspective “charged with the religious and social meanings it had for the early Renaissance” (p. 2). Why? “Renaissance artists and writers saw many techniques where we see a single discovery” (p. 8). Renaissance artists thought of perspective as a series of rules and techniques for constructing images; modern art historians, in contrast, think of perspective on analogy with tracing a scene on a pane of glass. In contrast to the “object oriented” Renaissance notion of perspective, the modern concept is “space oriented.” For the Renaissance, perspective was “a strategy for making pictures,” but for us it has become “a sign signifying a mental state, a culture, or an expressive language” (p. 17). Renaissance perspective originated as “a construction, an intellectual accomplishment rather than as a transcription of appearances” or a “revelation of optical reality” (pp. 136–38). To its contemporaries, Renaissance perspective suggested “the complicated, unfinished, and partial” rather than “the harmonious, balanced, and whole” (p. 170). We expect a uniform accuracy of perspective through an entire painting, but Renaissance practitioners were selective in their applications of perspective to passages of particular interest or significance. All these differences taken together, Elkins says, imply that our insistence on attributing to Renaissance paintings an isotropic application of perspective based on an ideal geometry is a mistake: “we are looking for something the Renaissance artists did not supply” (p. 226).Elkins’s excellent book is a corrective for the sort of thing Lillian Schwartz does in the April 1995 Scientific American. In the middle of a series of ill-substantiated conjectures, based on computer manipulations of images and ranging from the claim that Leonardo himself was a model for the Mona Lisa to the assertion that Piero della Francesca’s Resurrection of Christ, plastered over during the seventeenth century, must have been painted “in sunset colors,” she suggests that Leonardo intended The Last Supper to “appear as an extension of the refectory” in which it was painted. She explains the quirks of the painting’s perspective (“Leonardo placed Christ and the apostles up front, tilted the floor and table, designed side walls of uneven lengths and tapestries of different sizes and spacing”) by constructing a computer simulation to show that a person entering the room from what was then the main door “would perceive the supper as taking place in the refectory.” Schwartz’s conjecture may or may not be right, but comparison with her article reveals the major strengths of Elkins’s book. In contrast to Schwartz’s hasty manipulations, The Poetics of Perspective is encyclopedic; it takes scrupulous care to meet on their own ground the many [End Page 369] texts and paintings it examines, and it follows the Wittgensteinian imperative to stop thinking and look. Whatever the promise of the new technologies available to art historians, they can never replace the sort of patient attention Elkins lavishes on the works he studies.Harvey L. HixKansas City Art InstituteCopyright... (shrink)

This deeply researched, carefully constructed and very thoughtful book is fascinating in its own right as well as being indispensable background material for anyone interested in current philosophical thought about space, time, and geometry.

In 1824 Sadi Carnot published Réflexions sur la Puissance Motrice du Feu in which he founded almost the entire thermodynamics theory. Two years after his death, his friend Clapeyron introduced the famous diagram PV for analytically representing the famous Carnot’s cycle: one of the main and crucial ideas presented by Carnot in his booklet. Twenty-five years later, in order to achieve the modern version of the theory, Kelvin and Clausius had to reject the caloric hypothesis, which had influenced a few (...) of Carnot’s arguments. Relying on the possibility of studying the history of science by means of logical investigation, in this paper I shall propose an historical/epistemological research on Sadi Carnot’s original thermodynamics theory in which the French scientist presents more than two principles, all of which are expressed by double negated sentences (generally speaking) within non-classical logic. (shrink)

We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π01 class P is a subshift if and only if there exists a computable function F mapping 2ℕ to 2ℕ such that P is the set of itineraries of elements of 2ℕ. Π01 subshifts are constructed in 2ℕ and in 2ℤ which have no computable elements. We also consider the symbolic dynamics of maps on the unit interval.

I desire to invite the attention of students of Plato and of Greek mathematics to a solution of a passage which has long been a field of controversy for critics. For brevity's sake I shall take for granted an acquaintance with the two solutions which at present dispute the field, and further adopt certain positions which previous enquirers have established beyond reasonable doubt.

This provocative study revives a classical idea about rationality by developing analogies between the structure of personality and the structure of society in the context of contemporary work in the philosophy of mind, ehtics, decision theory, and social choice theory.

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories (...) as projective geometry; and not only as they were practiced in Kant’s time, but also as architects use them today. (shrink)

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be (...) class='Hi'>constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, (...) but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.The corresponding particle theory is independent of particle number, noninvariant under time reversal (arrow of time), compatible with quantum mechanics and has a corresponding positive definite canonical Hamiltonian associated with the clock of the source.We also provide a brief review of our work on the foundational aspects of the corresponding relativistic quantum theory. Here, we show that the standard square-root and Dirac equations are actually two distinct spin- $\frac{1}{2}$ particle equations. (shrink)

The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.

The concept of sustainable development considers environmental, social and economic issues in general. And the goals of resource conservation and socio-economic development do not contradict each other, but contribute to mutual reinforcement. The purpose of this study is to build and test an economic and mathematical model for the formation of strategies for the behavior of an economic entity with an increase in the impact of negative environmental factors. The proposed strategies and their models are based on the income-expenditure balance (...) equation, which takes into account both quantitative and qualitative characteristics. The constructed models are considered in the state space. The research methodology is based on building models in the form of linear combinations of functions of a homogeneous external impact and various spatial combinations of economic sources (sinks). The study makes it possible to assess the dependence of the amount of resources used for life support on the chosen adaptive strategy. Within the framework of the proposed model, it was found that the criterion for the effectiveness of the applied strategy can be an indicator of satisfaction with the state, the preservation of which, simultaneously with the preservation of the size of resources used, corresponds to the direction of optimization. This approach is consistent with the concept of sustainable development. (shrink)

This essay presents and analyzes the recent work of four prominent contemporary Jewish ethicists: Eugene Borowitz, David Novak, Byron Sherwin, and Walter Wurzburger. These authors are united in their affirmation of covenant as the central category of Jewish moral obligation and their concern to construct a Jewish ethic out of the classical sources of Judaism. Yet, as an individual analysis of their books will show, they adopt markedly different views of the authority of traditional Jewish law , the respective (...) roles of individual and community in moral deliberation, and the degree to which changing historical circumstances alter moral truths. (shrink)

This book offers insights relevant to modern history and epistemology of physics, mathematics and, indeed, to all the sciences and engineering disciplines emerging of 19th century. This research volume is the first of a set of three Springer books on Lazare Nicolas Marguérite Carnot’s (1753–1823) remarkable work: Essay on Machines in General (Essai sur les machines en général [1783] 1786). The other two forthcoming volumes are: Principes fondamentaux de l’équilibre et du mouvement (1803) and Géométrie de position (1803). Lazare Carnot (...) – l'organisateur de la victoire – in Essai sur le machine en général (1786) assumed that the generalization of machines was a necessity for society and its economic development. Subsequently, his new coming science applied to machines attracted considerable interest for technician, as well, already in the 1780’s. With no lack in rigour, Carnot used geometric and trigonometric rather than algebraic arguments, and usually went on to explain in words what the formulae contained. His main physical– mathematical concepts were the Geometric motion and Moment of activity–concept of Work . In particular, he found the invariants of the transmission of motion (by stating the principle of the moment of the quantity of motion) and theorized the condition of the maximum efficiency of mechanical machines (i.e., principle of continuity in the transmission of power). While the core theme remains the theories and historical studies of the text, the book contains an extensive Introduction and an accurate critical English Translation – including the parallel text edition and substantive critical/explicative notes – of Essai sur les machines en général (1786). The authors offer much-needed insight into the relation between mechanics, mathematics and engineering from a conceptual, empirical and methodological, and universalis point of view. As a cutting–edge writing by leading authorities on the history of physics and mathematics, and epistemological aspects, it appeals to historians, epistemologist–philosophers and scientists (physicists, mathematicians and applied sciences and technology). (shrink)

Background: The search for genetic variants between racial/ethnic groups to explain differential disease susceptibility and drug response has provoked sharp criticisms, challenging the appropriateness of using race/ethnicity as a variable in genetics research, because such categories are social constructs and not biological classifications.Objectives: To gain insight into how a group of genetic scientists conceptualise and use racial/ethnic variables in their work and their strategies for managing the ethical issues and consequences of this practice.Methods: In-depth semi-structured interviews were conducted with a (...) purposive sample of 30 genetic researchers who use racial/ethnic variables in their research. Standard qualitative methods of content analysis were used.Results: Most of the genetic researchers viewed racial/ethnic variables as arbitrary and very poorly defined, and in turn as scientifically inadequate. However, most defended their use, describing them as useful proxy variables on a road to “imminent medical progress”. None had developed overt strategies for addressing these inadequacies, with many instead asserting that science will inevitably correct itself and saying that meanwhile researchers should “be careful” in the language chosen for reporting findings.Conclusions: While the legitimacy and consequences of using racial/ethnic variables in genetics research has been widely criticised, ethical oversight is left to genetic researchers themselves. Given the general vagueness and imprecision we found amongst these researchers regarding their use of these variables, they do not seem well equipped for such an undertaking. It would seem imperative that research ethicist move forward to develop specific policies and practices to assure the scientific integrity of genetic research on biological differences between population groups. (shrink)

At the turn of the century, Gottlob Frege and Edmund Husserl both participated in the discussion concerning the foundations of logic and mathematics. Since the 1960s, comparisons have been made between Frege's semantic views and Husserl's theory of intentional acts. In quite recent years, new approaches to the two philosophers' views have appeared. This collection of articles opens with the first English translation of Dagfinn Føllesdal's early classic on Husserl and Frege of 1958. The book brings together a number of (...) new contributions by well-known authors and gives a survey of recent developments in the field. It shows that Husserl's thought is coming to occupy a central role in the philosophy of logic and mathematics, as well as in the philosophy of mind and cognitive science. The work is primarily meant for philosophers, especially for those working on the problems of language, logic, mathematics, and mind. It can also be used as a textbook in advanced courses in philosophy. (shrink)

Moral reasoning traditionally distinguishes two types of evil: moral and natural. The standard view is that ME is the product of human agency and so includes phenomena such as war, torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...) a new class of interesting and important examples of hybrid evil has come to light. In this paper, it is called artificial evil and a case is made for considering it to complement ME and NE to produce a more adequate taxonomy. By isolating the features that have led to the appearance of AE, cyberspace is characterised as a self-contained environment that forms the essential component in any foundation of the emerging field of Computer Ethics. It is argued that this goes some way towards providing a methodological explanation of why cyberspace is central to so many of CE’s concerns; and it is shown how notions of good and evil can be formulated in cyberspace. Of considerable interest is how the propensity for an agent’s action to be morally good or evil can be determined even in the absence of biologically sentient participants and thus allows artificial agents not only to perpetrate evil but conversely to ‘receive’ or ‘suffer from’ it. The thesis defended is that the notion of entropy structure, which encapsulates human value judgement concerning cyberspace in a formal mathematical definition, is sufficient to achieve this purpose and, moreover, that the concept of AE can be determined formally, by mathematical methods. A consequence of this approach is that the debate on whether CE should be considered unique, and hence developed as a Macroethics, may be viewed, constructively, in an alternative manner. The case is made that whilst CE issues are not uncontroversially unique, they are sufficiently novel to render inadequate the approach of standard Macroethics such as Utilitarianism and Deontologism and hence to prompt the search for a robust ethical theory that can deal with them successfully. The name Information Ethics is proposed for that theory. It is argued that the uniqueness of IE is justified by its being non-biologically biased and patient-oriented: IE is an Environmental Macroethics based on the concept of data entity rather than life. It follows that the novelty of CE issues such as AE can be appreciated properly because IE provides a new perspective. In light of the discussion provided in this paper, it is concluded that Computer Ethics is worthy of independent study because it requires its own application-specific knowledge and is capable of supporting a methodological foundation, Information Ethics. (shrink)

If α and β are ordinals, α ≤ β, and $\beta \nleq \alpha$ , then α + 1 ≤ β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA 0 , a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA (...) 0 and an arithmetical transfinite induction scheme. (shrink)

We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve (...) Markov's principle and the completeness of metric spaces. (shrink)

Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and differential equations. (...) This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of connexes and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s line elements and surface elements are discussed here in some detail. (shrink)

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some basic terms. Second, I (...) explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics. (shrink)

Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I (...) am concerned with the meaning of the set/class distinction and I wish to show that it arises naturally due to the nature of the sort of distinctions that sets create. Method: The method of the paper is to discuss first the Russell paradox and the arguments of Cantor that preceded it. Then we point out that the Russell set of all sets that are not members of themselves can be replaced by the Russell operator R, which is applied to a set S to form R(S), the set of all sets in S that are not members of themselves. Results: The key point about R(S) is that it is well-defined in terms of S, and R(S) cannot be a member of S. Thus any set, even the simplest one, is incomplete. This provides the solution to the problem that I have posed. It shows that the distinction between sets and classes is natural and necessary. Implications: While we have shown that the distinction between sets and classes is natural and necessary, this can only be the beginning from the point of view of epistemology. It is we who will create further distinctions. And it is up to us to maintain these distinctions, or to allow them to coalesce. Constructivist content: I argue in favor of a constructivist perspective for set theory, mathematics, and the way these structures fit into our natural language and constructed speech and worlds. That is the point of this paper. It is only in the reach for absolutes, ignoring the fact that we are the authors of these structures, that the paradoxes arise. (shrink)

Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In _Oppositions and Paradoxes_, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not (...) only motivates abstract conceptual thinking about the paradoxes at issue, but he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life — he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or think of an office mailbox labelled “mail for those with no mailbox”—if this is a person’s mailbox, how can they possibly have “no mailbox”? These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts. (shrink)

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA ${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and (...) the Dialectica interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis. (shrink)

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 thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

To localize wheat ESTs on chromosomes, 882 homoeologous group 6-specific ESTs were identified by physically mapping 7965 singletons from 37 cDNA libraries on 146 chromosome, arm, and sub-arm aneuploid and deletion stocks. The 882 ESTs were physically mapped to 25 regions flanked by 23 deletion breakpoints. Of the 5154 restriction fragments detected by 882 ESTs, 2043 were localized to group 6 chromosomes and 806 were mapped on other chromosome groups. The number of loci mapped was greatest on chromosome 6B and (...) least on 6D. The 264 ESTs that detected orthologous loci on all three homoeologs using one restriction enzyme were used to construct a consensus physical map. The physical distribution of ESTs was uneven on chromosomes with a tendency toward higher densities in the distal halves of chromosome arms. About 43% of the wheat group 6 ESTs identified rice homologs upon comparisons of genome sequences. Fifty-eight percent of these ESTs were present on rice chromosome 2 and the remaining were on other rice chromosomes. Even within the group 6 bins, rice chromosomal blocks identified by 1-6 wheat ESTs were homologous to up to 11 rice chromosomes. These rice-block contigs were used to resolve the order of wheat ESTs within each bin. (shrink)

This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely (...) the duality of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. (shrink)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his (...) power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (shrink)