We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate (...) its utility in deciding when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations. We also give a generic backward error based method that is equivalent to iteration for regular perturbation. (shrink)

Interest in the computational aspects of modeling has been steadily growing in philosophy of science. This paper aims to advance the discussion by articulating the way in which modeling and computational errors are related and by explaining the significance of error management strategies for the rational reconstruction of scientific practice. To this end, we first characterize the role and nature of modeling error in relation to a recipe for model construction known as Euler’s recipe. We then describe a (...) general model that allows us to assess the quality of numerical solutions in terms of measures of computational errors that are completely interpretable in terms of modeling error. Finally, we emphasize that this type of error analysis involves forms of perturbation analysis that go beyond the basic model-theoretical and statistical/probabilistic tools typically used to characterize the scientific method; this demands that we revise and complement our reconstructive toolbox in a way that can affect our normative image of science. (shrink)

In this book a comprehensive study of the theories of error in classical Indian philosophy has been made by applying the techniques of linguistic and conceptual analysis. The special feature of this volume is that the different tools of analysis are directly applied on the classical Indian arguments. By way of analysis, it is maintained that the Indian theories of error are neither advancing a psychological account of error nor are the theories advancing any (...) metaphysical theory regarding the ontological status of the object of error but are primarily engaged in unravelling the logical structure of the concept of error. It is held in this work that the Khyativadas are theories about the logical nature or erroneous about the ontological status of the object. (shrink)

This study aimed to investigate the longitudinal trends in mathematical competencies of Grade 8 students in a public high school located in Zamboanga del Sur, Philippines. The study collected data over a period of six academic years, allowing for a comprehensive analysis of students' performance in 16 distinct mathematical competences of basic education curriculum. These topics include, but are not limited to, special products and factors, factoring, and basic concepts of probability. Using a quantitative research design, the study analyzed (...) both numerical and secondary data to identify patterns or trends in students' mathematical competencies. Trends were analyzed through regression analysis using slopes. The significance of these trends was then evaluated using t-tests for the standard error of slopes. The statistical analysis revealed significant upward trends in certain mathematical topics, indicating an improvement in the students' competencies over the six academic years. However, certain topics displayed downward or stable trends, indicating areas for improvement in the teaching methods used for these topics. The study highlights the importance of flexible and adaptable approaches to mathematics education. It emphasizes the need for constant evaluation and improvement of teaching methods to enhance student learning outcomes. The findings of this study provide valuable insights for educators and policymakers to guide evidence-based decision-making and improve the quality of mathematics education in the new normal setup. Implications and future research directions are incorporated in the study. (shrink)

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims that "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Contrary to this view, this thesis (...) delineates and implements a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified. (shrink)

In his philosophical history of nineteenth-century mathematics, Proofs and Persuasions: The Logic of Mathematical Discovery, Imre Lakatos asserts that mathematical criticism was the driving force in the growth of mathematical knowledge during the nineteenth century, and provided the impetus for some of the deepest conceptual reformulations of the century. The philosophy of mathematics represented by Proofs and Refutations also presents a rich analysis of how mathematics can be thought of as an essentially historical discipline. Despite protestations (...) by Lakatos that he completely discarded Hegel when he discovered the work of Karl Popper, his philosophy of error closely resembles Hegel's in the Phenomenology of Spirit. According to Lakatos, the impact of proofs and refutations on naive concepts is to erase them completely and replace them by proof-generated concepts. His historical point about proofs and refutations is that this pattern in the growth of mathematical knowledge is a relatively recent innovation. Hegel's and Lakatos' shared vision of theoretical knowledge is that rather than being the inspired work of timeless, totally subjective, intellectual intuition , it is, more often than not, mediated by a lengthy history of speculation and failure. (shrink)

ACMES is a multidisciplinary conference series that focuses on epistemological and mathematical issues relating to computation in modern science. This volume includes a selection of papers presented at the 2015 and 2016 conferences held at Western University that provide an interdisciplinary outlook on modern applied mathematics that draws from theory and practice, and situates it in proper context. These papers come from leading mathematicians, computational scientists, and philosophers of science, and cover a broad collection of mathematical and philosophical topics, (...) including numerical analysis and its underlying philosophy, computer algebra, reliability and uncertainty quantification, computation and complexity theory, combinatorics, error analysis, perturbation theory, experimental mathematics, scientific epistemology, and foundations of mathematics. By bringing together contributions from researchers who approach the mathematical sciences from different perspectives, the volume will further readers' understanding of the multifaceted role of mathematics in modern science, informed by the state of the art in mathematics, scientific computing, and current modeling techniques. (shrink)

In this paper, a mathematical model of the respiratory mechanics is used to reproduce experimental signal waveforms acquired from three newborn lambs. As the main challenge is to determine specific lamb parameters, a sensitivity analysis has been realized to find the most influent parameters, which are identified using an evolutionary algorithm. Results show a close match between experimental and simulated pressure and flow waveforms obtained during spontaneous ventilation and pleural pressure variations acquired during the application of positive pressure, since (...) root mean square errors equal to 0.0119, 0.0052 and 0.0094. The identified parameters were discussed in light of previous knowledge of respiratory mechanics in the newborn. (shrink)

We report on an unpublished and previously unknown manuscript of John von Neumann and contextualize it within the development of the theory of shock waves and detonations during the nineteenth and twentieth centuries. Von Neumann studies bombs comprising a primary explosive charge along with explosive booster material. His goal is to calculate the minimal amount of booster needed to create a sustainable detonation, presumably because booster material is often more expensive and more volatile. In service of this goal, he formulates (...) and analyzes a partial differential equation-based model describing a moving shock wave at the interface of detonated and undetonated material. We provide a complete transcription of von Neumann’s work and give our own accompanying explanations and analyses, including the correction of two small errors in his calculations. Today, detonations are typically modeled using a combination of experimental results and numerical simulations particular to the shape and materials of the explosive, as the complex three dimensional dynamics of detonations are analytically intractable. Although von Neumann’s manuscript will not revolutionize our modern understanding of detonations, the document is a valuable historical record of the state of hydrodynamics research during and after World War II. (shrink)

This is the story, told in the light of a new analysis of historical data, of a mathematical biology problem that was explored in the 1930s in Thomas Morgan’s laboratory at the California Institute of Technology. It is one of the early developments of evolutionary genetics and quantitative phylogeny, and deals with the identification and counting of chromosomal inversions in Drosophila species from comparisons of genetic maps. A re-analysis of the data produced in the 1930s using current (...) class='Hi'>mathematics and computational technologies reveals how a team of biologists, with the help of a renowned mathematician and against their first intuition, came to an erroneous conclusion regarding the presence of phylogenetic signals in gene arrangements. This example illustrates two different aspects of a same piece: the appearance of a mathematical in biology problem solved with the development of a combinatorial algorithm, which was unusual at the time, and the role of errors in scientific activity. Also underlying is the possible influence of computational complexity in understanding the directions of research in biology. (shrink)

Despite his unreserved appreciation of Turing’s analysis for being a “precise and unquestionably adequate definition” of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a “philosophical error” in Turing’s argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing’s work, despite the error of its ways? Previous interpretative strategies by Feferman, (...) Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing’s response to the “mathematical objection” based on Gödel’s incompleteness theorem and Gödel’s own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no “error” on Turing’s side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together. (shrink)

In his late fragment, ‘Sources of Knowledge of Mathematics and Natural Sciences’ Frege laments the tendency to confuse functions with objects and says, ‘It is here that the tendency of language by its use of the definite article to stamp as an object what is a function and hence a non-object, proves itself to be the source of inaccurate and misleading expressions and also of errors of thought. Probably most of the impurities that contaminate the logical source of knowledge (...) have their origins in this.’ This paper applies Frege’s logical insights to a series of influential modal arguments for the existence of souls or minds, in order to demonstrate that they hinge on the error of treating functions as objects. Related arguments have also resulted in the claim that identity statements are necessary, and that apparent identities ‘really’ involve relations of constitution. Here it is argued that once we recognise that all these arguments involve the assimilation of objects and concepts (which Frege identifies with functions) they can be seen to be defective. (shrink)

My life as a quantum physicist / M. Nakahara -- A review on operator quantum error correction - Dedicated to Professor Mikio Nakahara on the occasion of his 60th birthday / C.-K. Li, Y.-T. Poon and N.-S. Sze -- Implementing measurement operators in linear optical and solid-state qubits / Y. Ota, S. Ashhab and F. Nori -- Fast and accurate simulation of quantum computing by multi-precision MPS: Recent development / A. Saitoh -- Entanglement properties of a quantum lattice-gas model (...) on square and triangular ladders / S. Tanaka, R. Tamura and H. Katsura -- On signal amplification from weak-value amplification / Y. Shikano -- Topological protection of quantum information / K. Fujii -- Quantum annealing with antiferromagnetic fluctuations for mean-field models / Y. Seki and H. Nishimori -- A method to change phase transition nature - Toward annealing methods / R. Tamura and S. Tanaka -- Computational analysis of the first stage of the photosynthetic system, the light-dependent reaction, by quantum chemical simulation method / M. Tada-Umezaki -- Two-qubit gate operation on selected nearest neighboring qubits in a neutral atom quantum computer / E. Hosseini Lapasar... [et al.] -- A simple operator quantum error correction scheme avoiding fully correlated errors / C. Bagnasco, Y. Kondo and M. Nakahara -- Black hole predictability, classical and quantum / A. Ishibashi -- Classical field simulation of finite-temperature Bose gases / T. Sato -- Atomic quantum simulations of lattice gauge theory: Effect of gauge symmetry breaking / K. Kasamatsu, I. Ichinose and T. Matsui -- Recursive construction of noiseless subsystem for qudits / U. Gungordu... [et al.] -- Composite quantum gates for precise quantum control / M. Bando... [et al.] -- New formulation of statistical mechanics using thermal pure quantum states / S. Sugiura and A. Shimizu -- Thermodynamics in unitary time evolution / T. N. Ikeda -- Second law of thermodynamics with QC-mutual information / T. Sagawa. (shrink)

With the widespread application of information technology and the continuous deepening of my country’s education informatization construction, the educational field has increasingly higher requirements for teachers’ informatization quality. The advancement of the process of education informatization depends to a large extent on the level of teachers’ own level of informatization and has gradually become an important indicator to measure the informatization of education. The main body of education and the decisive factor are teachers. The scientific and reasonable evaluation of informatization (...) education is an important content of today’s education informatization. This paper uses regression analysis to evaluate the teacher education information system and uses mathematical statistics to establish a linear regression model for the city and analyze and calculate it. Through the quantitative analysis of the evaluation index system of the education informatization level, the multiple linear regression analysis method was used to establish the linear regression mathematical model of the school education informatization level on each secondary index, and the influence of various indicators on the school education informatization level was analyzed to establish a scientific evaluation system to evaluate the literacy of teachers’ informatization education and to give suggestions for improving the level of school education informatization. The design of the evaluation system is more scientific and reasonable, the regression error is small, the results obtained are more accurate, and the impact of nonlinear errors is basically eliminated. (shrink)

This paper focuses on the theme of the application of deep learning in the field of basketball sports, using research methods such as literature research, video analysis, comparative research, and mathematical statistics to explore deep learning in real-time analysis of basketball sports data. The basketball posture action recognition and analysis system proposed for basketball movement is composed of two parts serially. The first part is based on the bottom-up posture estimation method to locate the joint points and (...) is used to extract the posture sequence of the target in the video. The second part is the analysis and research of the action recognition algorithm based on the convolution of the space-time graph. According to the extracted posture sequence, the basketball action of the set classification is recognized. In order to obtain more accurate and three-dimensional information, a multitraining target method can be used in training; that is, multiple indicators can be detected and feedback is provided at the same time to correct player errors in time; the other is an auxiliary method, which is compared with ordinary training. The method can actively correct technical movements, train players to form muscle memory, and improve their abilities. Through the research of this article, it provides a theoretical basis for promoting the application of deep learning in the field of basketball and also provides a theoretical reference for the wider application of deep learning in the field of sports. At the same time, the designed real-time analysis system of basketball data also provides more actual reference values for coaches and athletes. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 1-12 [Access article in PDF] Art and Mathematics in Education Richard Hickman and Peter Huckstep We begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, (...) on the other, work in art that has a mathematical basis. Observations of these kinds are not trivial. They have significant implications for the teaching of these areas of the curriculum in at least two ways. First, there is the methodological issue of the extent to which we should teach mathematics and art separately, and second, the teleological question of why they appear in the curriculum at all. So the relationship between the nature of mathematics and art, perceived or real, bears down on questions of the individuation and the justification of these disciplines, or, in other words, upon pedagogy and purpose. Although in principle both the pedagogy and the purpose of any discipline are distinct, there are important connections between them, as we shall draw out.As far as mathematics goes, it has been stressed that the purposes of even rudimentary aspects of mathematics such as counting do need to be made explicit to pupils. 1 What have been seen as errors in understanding are internally linked with pupils' ideas of the purposes of counting. Anna Sierpinska discusses a similar situation within the broader aspect of what she calls "theory" in higher mathematics. 2 More generally, John Passmore, in his analysis of pupils' understanding includes the case where a pupil sees no need to understand since he or she cannot understand the point of learning the subject. 3 If the purposes of art and mathematics are strikingly similar then there is reason to suppose that certain pedagogical approaches that [End Page 1] follow from this will improve understanding of both mathematics and art, and therefore understanding of reasons for engaging in math-based and art-based activities.In schools we see what appears to be some evidence of connections between art and mathematics in cross-curricular work, such as that described in Jacqueline Cossentino and David Schaffer. 4 Apart from providing much-needed motivation, an important value of this approach toward mathematics is that it can illuminate pupils' understanding of some of its purpose by, at the very least, allowing them to recognize instances of mathematics at work, often in unsuspecting contexts. In particular, by facilitating some transference of knowledge pupils can appreciate the so-called "power" of mathematics in its widespread application to the world of artefacts produced by themselves and others.Can we go further than displaying connections between these disciplines? Can we treat mathematics itself as art? In particular, can we suppose that pupils can make mathematics just as they can make artefacts? Some writers seem to imply this by invoking the notion of creativity as a common property between the two disciplines. 5 Leaving aside the ironic sense of creativity in, for example, the case of "creative accountancy" pupils must indeed "make" sense of mathematics and they must originate mathematical assertions of their own that have not been directly taught. In both respects mathematics is similar to the learning of a language. Further than this, we value pupils' inventiveness in their approaches to areas of mathematics, calculations for example.The linking of mathematics and art via creativity has its most persuasive force in the change of attitude toward axiomatic foundations. 6 The story is familiar: for centuries Euclidean geometry was (wrongly) supposed to offer truths of the actual space in which we lived, since its axioms were thought to be confirmed by our intuitions of space. However, when it was found that consistent and useful axiomatic systems could be constructed by making certain alterations to these axioms, it became clear that mathematics was not simply discovered. It involved a significant measure of creativity. Yet one thing is clear: the sense that children make of mathematics, the original utterances they make... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52 Reviews  HISTORY OF PHILOSOPHICAL ANALYSIS C P Philosophy / Purdue U. West Lafayette,  ,  @. Scott Soames. Philosophical Analysis in the Twentieth Century, Vol. : The Dawn of Analysis; Vol. : The Age of Meaning. Princeton: Princeton U. P., . Pp. xix, ; xxii, . . (hb), . (pb) for each volume. he last twenty (...) years have seen an explosion in books and papers on RusTsell ’s philosophy and its contemporary significance. There is good reason to think that this will continue as the contents of the Collected Papers are digested by Russell scholars and as more specialists contribute to the history of analytic philosophy more generally. Given all this good news, it is disconcerting to find a -page discussion of Russell, in a well-reviewed book by a first-rate philosopher, repeating many of the errors and misconceptions about Russell that scholars have worked so hard against. Soames’ discussion of Russell in the volumes under review is in fact so distressing that it alone compromises the book as a suitable introduction to the history of analytic philosophy. After briefly reviewing the outline of the two volumes, I discuss the errors concerning Russell, and conclude by drawing some lessons for Russell scholarship. Soames’ focus is on what he takes to be the most important and influential work of analytic philosophers, beginning with Moore’s Principia Ethica and ending in  with Kripke’s Naming and Necessity lectures. Kripke, in fact, marks the culmination of one of the two great achievements of analytic philos- _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52  Reviews ophy that Soames sees in this period: … the two most important achievements that have emerged from the analytic tradition in this period are (i) the recognition that philosophical speculation must be grounded in pre-philosophical thought, and (ii) the success achieved in understanding, and separating one from another, the fundamental methodological notions of logical consequence, logical truth, necessary truth, and apriori truth. (: xi) Moore is credited with the methodological innovation required by (i), as once we accept that what we think we know prior to philosophical reflection is a constraint on our epistemology, Moore’s response to scepticism, which Soames endorses, inevitably follows. But even Moore, and nearly every other figure that Soames discusses, is guilty of confusing necessity, analyticity and apriority. Soames endorses Kripke’s basic point that necessity is a metaphysical concept that can come apart from the epistemic notion of apriority and the semantic category of analytic propositions. Volume  repeats this charge several times, using it to undermine Moore’s views on ethics in Part One, Russell’s conception of analysis in Part Two, and logical positivism in Part Three. It is noteworthy that Soames takes Ayer’s Language, Truth and Logic as representative of logical positivism, ignoring contemporary scholarship on the Vienna Circle just as much as he ignores Russell scholarship. Part Four reconstructs Wittgenstein’s views in the Tractatus and Volume  ends with a discussion of Quine’s “Two Dogmas of Empiricism”. The second volume begins with a part on Wittgenstein’s Philosophical Investigations. This paves the way for Soames’ discussion of ordinary language philosophy in Parts Two and Three, which Soames sees as closely tied to the later Wittgenstein. Ryle’s Concept of Mind, Strawson’s early views on truth, Hare’s theory of goodness as well as Malcolm’s paradigm-case argument and Austin’s Sense and Sensibilia are investigated in these chapters. Part Four presents Grice’s theory of conversational implicature as the final nail in the coffin of ordinary language philosophy. In Part Five, Soames returns to Quine, this time discussing the ambitious arguments of Word and Object and the more general project of naturalized epistemology. Part Six articulates Davidson’s program for constructing a theory of meaning for natural languages along the lines of a Tarskistyle theory of truth, and Volume  ends with an extended discussion of the promise and limitations of Kripke’s Naming and Necessity. The material on Russell is entirely confined to four chapters in Volume : Chapter : “Logical Form, Grammatical Form, and the Theory of Descriptions ”, Chapter : “Logic and Mathematics: The Logicist Reduction”, Chapter : “Logical Constructions and... (shrink)

This article argues that many, if not most, behavior descriptions and sequencing are in essence an interpretation of signs, and are evaluated as sequences of signs by researchers. Thus, narrative analysis, as developed by Barthes and others, seems best suited to be used in behavioral/biosemiotic studies rather than mathematical modeling, and is very similar to some classic ethology methods. As our brain interprets behaviors as signs and attributes meaning to them, narrative analysis seems more suitable than mathematical modeling (...) to describe and study behavior. Mathematical models are, on many occasions, extremely reductionist and simplifying because of computational and/or numerical representation limitations that lead to errors and straitjacketing interpretations of reality. Since actual animals are not as optimal in real life as our mathematical models, here it is proposed that we should consider logical/verbal models and semantic interpretations as equally or even better suited for behavioral analysis, and refrain from enforcing mathematical modeling as the only way to study and understand biological problems, especially those of a behavioral and biosemiotic nature. (shrink)

This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay (...) subject matter limit function with the number of students by 16 (sixteen). Data collection techniques used are shaped test methods essay, interview method to students who have misconceptions, and methods of documentation of the test answers. Examination of the validity of the data using a triangulation technique that compares the data written test results with data from interviews. The results of this study can be described as follows; (1) The level of misconceptions experienced by students belonging to the category of low, amounting to 12.18%. However, students who do not understand the concept quite high at 40.38%, and the others are students who understand the concept that is equal to 47.44%. (2) The misconception most experienced students lie in subconcepts explain the meaning of limit function at one point through the calculation of values around that point, that is equal to 20.51%. The tendency misconceptions experienced by students is located on errors and operating concepts that misconceptions students that there should be no limit in the completion of the writing of the emblem and misconceptions about the limit students to conclude if the limit value of 0 is no limit to the value of the function. (shrink)

Lisciandra poses a challenge for robustness analysis as applied to economic models. She argues that substituting tractability assumptions risks altering the main mathematical structure of the model, thereby preventing the possibility of meaningfully evaluating the same model under different assumptions. In such cases RA is argued to be inapplicable. However, Lisciandra is mistaken to take the goal of RA as keeping the mathematical properties of tractability assumptions intact. Instead, RA really aims to keep the modeling component while varying the (...) corresponding mathematical formulation. Thus, her argument concerning whether the associated mathematical properties of certain assumptions can be kept intact is irrelevant to the success of RA. Furthermore, we explicate and develop Lloyd’s account of “model robustness” to provide solutions to Lisciandra’s challenges. Our solutions are, namely, error analysis and independent empirical support. We conclude that although complex economic models do face potential dangers, there are solutions and robustness analysis need not be given up.. (shrink)

The relatively recent increase in prominence of computer simulations in scientific inquiry gives us more reasons than ever before for asserting that mathematics is a wonderful tool. In fact, a practical knowledge of scientific computation has become essential for scientists working in all disciplines involving mathematics. Despite their incontestable success, it must be emphasized that the numerical methods subtending simulations provide at best approximate solutions and that they can also return very misleading results. Accordingly, epistemological sobriety demands that (...) we clarify the circumstances under which simulations can be relied upon. With this in mind, this paper articulates a general perspective to better understand and compare the strengths and weaknesses of various error-analysis methods. (shrink)

The Bristol Inquiry is arguably one of the most important cases of judicial medical investigation held in the United Kingdom, which continues to raise important insights into the social construction of medical and scientific risks. As a way of marking the inquiry’s tenth anniversary year, this article returns to an important conversation held between noted pediatric cardiothoracic and cardiovascular specialists, on days 49 and 50 of the inquiry’s proceedings. Their conversance principally describes a pathway of scientific advancement across four decades. (...) Risks, and the avoidance of medical error, represent important subtopics of the ensuing historical narrative, within which the experts describe opportunities for error as diminishing in-line with paradigmatic advancement. The telling of this story of risk mitigation involves the expurgation of considerable sensitivities to scientific uncertainty. In analyzing this example of expert scientific witness testimony, the article primarily considers how representations of uncertainty are formulated in language. The article concludes that, in this instance, the social construction of risk and uncertainty were heteroglossically premised upon language catalysts, cognitive metaphors, and logic-based mathematical rubrics. (shrink)

Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction (...) in numerical economics could mean. Method: An algebraic simple model system is subjected to a deeper structure of underlying variables. With an algorithm simulating the steps in taking a limit of second order difference quotients the error terms are studied at the background of their algebraic expression. Results: With the algorithm that was applied to a simple quadratic polynomial system we found unstably amplified round-off errors. The possibility of numerical chaos is already known but not in such a simple system as used in our paper. The amplification of the errors implies that it is not possible with computer means to constructively show that the algebra and numerical analysis will ‘on the long run’ converge to each other and the error term will vanish. The algebraic vanishing of the error term cannot be demonstrated with the use of the computer because the round-off errors are amplified. In philosophical terms, the amplification of the round-off error is equivalent to the continuum hypothesis. This means that the requirement of (numerical) construction of mathematical objects is no safeguard against inference-only conclusions of qualities of (numerical) mathematical objects. Unstably amplified round-off errors are a same type of problem as the ordering in size of transfinite cardinal numbers. The difference is that the former problem is created within the requirements of constructive mathematics. This can be seen as the reward for working numerically constructive. (shrink)

In this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelveEphemeridesand sevenProcedure Texts. Three computational systems of (...) Mercury are known, all of system A. System A1is represented by nineEphemeridescovering the years 190 BC to 100 BC and system A2by twoEphemeridescovering the years 310 to 290 BC. System A3is known from aProcedure Textand from Text M, anEphemerisof the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in theAstronomical DiariesandPlanetary Textswe find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase. (shrink)

“What data will show the truth?” is a fundamental question emerging early in any empirical investigation. From a statistical perspective, experimental design is the appropriate tool to address this question by ensuring control of the error rates of planned data analyses and of the ensuing decisions. From an epistemological standpoint, planned data analyses describe in mathematical and algorithmic terms a pre-specified mapping of observations into decisions. The value of exploratory data analyses is often less clear, resulting in confusion about (...) what characteristics of design and analysis are necessary for decision making and what may be useful to inspire new questions. This point is addressed here by illustrating the Popper-Miller theorem in plain terms and using a graphical support. Popper and Miller proved that probability estimates cannot generate hypotheses on behalf of investigators. Consistently with Popper-Miller, we show that probability estimation can only reduce uncertainty about the truth of a merely possible hypothesis. This fact clearly identifies exploratory analysis as one of the tools supporting a dynamic process of hypothesis generation and refinement which cannot be purely analytic. A clear understanding of these facts will enable stakeholders, mathematical modellers and data analysts to better engage on a level playing field when designing experiments and when interpreting the results of planned and exploratory data analyses. (shrink)

Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then (...) the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist. (shrink)

Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. -/- John Corcoran, Counterexamples and Proexamples. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 E-mail: [email protected] Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition “every perfect number is even” is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition that some (...) perfect number is odd. Conversely, every proexample for the proposition “some perfect number is odd” is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition “every true proposition is known to be true” is false–necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample–sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts. (shrink)

The paper outlines various concepts of rationality, their characteristics and consequences. In the first, most general part, the metaphysical, instrumental and discursive rationality is distinguished. The following part focuses on instrumental rationality and the rational choice theory and ordinal and cardinal utility, expected utility and game theory, respectively. All those concepts are summarised as being the most mathematically elegant and mostly decidable and helpful in the decision-making process. Giving primacy to individual preferences and withholding the judgment on their “objective” value, (...) they are also devoid of double standards. They are, however, strongly normative and weakly coincide with actual agents’ behaviour. Empirical findings on agents’ decision making seem to demonstrate their irrationality, unless we introduce into the analysis different concepts of rationality, namely based on costs efficient heuristics, inclusive fitness and ecological rationality. They are discussed respectively, and although they seem better to explain the set of humans’ seemingly irrational behaviour, they are likely week in predicting that behaviour. They are also losing their normative dimension and thus cease to be helpful in decision making. Applying the particular theory of rationality, either descriptively or normatively, seems to depend strongly on the environment, which can be characterised by its extension from a small to a large world. The more the small world’s features an environment reveals, the more effective is the application of the particular model of rationality. Beyond the small worlds, rule stochasticity, underspecification and misspecification and the only reasonable method are consecutive trials and errors, which eventually may reduce the large world to the small one. (shrink)

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response (e.g. Field’s program) accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft (...) reconstrual equivalent to standard mathematics. A second style of response (suggested, e.g., by Balaguer and Maddy) denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes (e.g., if the differential equation is linear) the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories (e.g., theory of differential equations) presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear (a) how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and (b) how this role is difficult to square with fictionalism. (shrink)

Critical thinking is a 21st century skill that allows students to think reflectively and reason in making decisions. Students' critical thinking influences success in solving numeracy problems. This study aims to explain the ability to think critically in solving numeracy problems with high self-efficacy. The study employed a qualitative approach and phenomenological method. Data collection involved the use of the Group Embedded Figures Test (GEFT), self-efficacy questionnaires, and numeracy assessments in mathematics, specifically on topics related to cubes and rectangular (...) prisms. Data analysis comprised data condensation, data presentation, and conclusion drawing. Based on GEFT and self-efficacy questionnaire results, one class V-B student with a field-dependent cognitive style and high self-efficacy was selected for further critical thinking skills assessment. Based on the results the student with a field-dependent cognitive style and high self-efficacy demonstrated notable strengths. In the process of basic clarification, the student met the indicators for analyzing arguments and formulating statements, effectively presenting their arguments. When providing reasons for decisions, the student showed the ability to consider the credibility of sources by selecting appropriate problem-solving strategies. However, she occasionally made calculation errors due to inaccuracy. In the process of concluding, the student met the indicators for making and evaluating decisions, successfully composing conclusion sentences that aligned with the problems and final answers. Students with field dependent cognitive style and high self efficacy are able to fulfill all indicators of critical thinking by using their ability to solve critical thinking problems. He excelled in verbal ability at the time of the interview by explaining the argument of the problem and the answer, but his weakness is less thorough in working so that the results obtained are less precise. (shrink)

The ArgumentWhile there has been muchattention given to experiment in modern science studies, there has been astoundingly little concern spared over the practice ofquanitataivemeasurment.Thus myths about the unreasonable effectiveness of mathematice in science still abound. This paper presents: An explicit mathematical model of the stabilization of quantitative constants in a mathematical science to rival older Bayesian and classical accounts;a framework for writing a history of pracitces with regard to treatment of quantitative measurement erroe; resourece for the comparative sociology of differing (...) discipliness in this regard;and a prolegonmena to a critique of orthodox economics and accounting theories. The key to all these diverse themes is the realization that no one individual alone is capable of fixing the magnitude of a quantitative error estimate, and therefore the social construction of error must be given a more precise meaning, an therefore the social construction of error must be given a more precise menaing, and that this occurs through the istrumentality of meta-analysis. (shrink)

The author analyses the reliability and accuracy of the results of sociological research conducted using quota sampling. The author discusses the advantages and disadvantages of using quota samples compared to random samples. The main focus is on the representativeness of the obtained data and the possibility of extrapolating them to the general population.In the problem statement, the author noted that a mechanical method of forming a random sample is most often used to study public opinion in Ukraine. The main feature (...) of this method is to ensure an equal probability of selection of each element of the general population. The author emphasizes that although this method allows for easy calculation of possible errors and analysis of collected information, it has a disadvantage in representing specific groups of the population.To address this problem, the author proposes using an equal probability method of forming quota samples, which allows selecting respondents according to their quotas in the general population, thus providing a more representative model of the studied population. However, in the scientific community, there is no consensus on the use of indicators in forming quota samples, and all existing works remain at the level of theoretical assumptions.The author attempts to empirically substantiate the effectiveness of quota samples through experimentation. The hypothesis of the experiment assumes that the results of studies conducted according to quotas can be extrapolated to the general population with high accuracy and reliability using mathematical and statistical methods. During the experiment, both quota and random samples were used to compare the results.The general population consisted of 489 musical compositions represented in different proportions to eliminate the subjective component from the experiment and provide the opportunity to clearly assess and record the parameters of the initial general population for further comparison with the results of different types of samples. The musical compositions had two parameters: the name of the performer for identification and the duration of the track for digital and statistical evaluation. The analysis was carried out using a specially developed program for statistical data analysis. The collected data allowed for the assessment of the average track duration, median, skewness, standard deviation, and other statistical indicators.The results of the experiment showed that the quota sample provides more accurate results when the quotas are followed during the selection of elements of the general population. The sample obtained by random selection without adhering to quotas showed greater variability in indicators such as mean, median, skewness, and standard deviation relative to the general population. Thus, it was concluded that quota samples have advantages in sociological research, especially when specific groups of the population need to be considered.The author emphasizes the need for further research and improvement of quota sampling methods to enhance the accuracy and reliability of sociological studies, which are important for examining particular social issues. (shrink)

Both Medawar and Hamilton contributed key ideas to the modern evolutionary theory of ageing. In particular, they both suggested that, in populations with overlapping generations, the force with which selection acts on traits declines with the age at which traits are expressed. This decline would eventually cause ageing to evolve. However, the biological literature diverges on the relationship between Medawar’s analysis of the force of selection and Hamilton’s. Some authors appear to believe that Hamilton perfected Medawar’s insightful, yet ultimately (...) erroneous analysis of this force, while others see Hamilton’s analysis as a coherent development of, or the obvious complement to Medawar’s. Here, the relationship between the two analyses is revisited. Two things are argued for. First, most of Medawar’s alleged errors that Hamilton would had rectified seem not to be there. The origin of these perceived errors appears to be in a misinterpretation of Medawar’s writings. Second, the mathematics of Medawar and that of Hamilton show a significant overlap. However, different meanings are attached to the same mathematical expression. Medawar put forth an expression for the selective force on age-specific fitness. Hamilton proposed a full spectrum of selective forces each operating on age-specific fitness components, i.e. mortality and fertility. One of Hamilton’s expressions, possibly his most important, is of the same form as Medawar’s expression. But Hamilton’s selective forces on age-specific fitness components do not add up to yield Medawar’s selective force on age-specific fitness. It is concluded that Hamilton’s analysis should be considered neither as a correction to Medawar’s analysis nor as its obvious complement. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations of mathematicians expended on refining the methods of (...) reasoning with formulae and predicate calculus. Next, two main groups of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained, methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several other problems constitute new, specifically diagrammatic ways of reliable reasoning. (shrink)

This paper is concerned with the event-based state and fault estimation problem for a class of linear discrete systems with randomly occurring faults and missing measurements. Different from the static event-based transmission mechanism with a constant threshold, a dynamic event-based mechanism is exploited here to regulate the threshold parameter, thus further reducing the amount of data transmission. Some mutually independent Bernoulli random variables are used to characterize the phenomena of ROFs and missing measurements. In order to simultaneously estimate the system (...) state and the fault signals, the main attention of this paper is paid to the design of recursive filter; for example, for all DETM, ROFs, and missing measurements, an upper bound for the estimation error covariance is ensured and the relevant filter gain matrix is designed by minimizing the obtained upper bound. Moreover, the rigorous mathematical analysis is carried out for the exponential boundedness of the estimation error. It is clear that the developed algorithms are dependent on the threshold parameters and the upper bound together with the probabilities of missing measurements and ROFs. Finally, a numerical example is provided to indicate the effectiveness of the presented estimation schemes. (shrink)

This paper responds briefly to the criticism of Rodrigues and Rosa on my earlier analysis of the twin paradox. The main point that I have emphasized (and that the authors have not refuted, either logically or mathematically) is the error in directly identifying anabstract measure relative to a reference frame [and its transformations to all other possible reference frames in which the laws of nature are to be compared (such as temporal and spatial measures)], with aphysical extension and (...) duration of a material body. (shrink)

This paper presents an analysis of the systematic astronomical observations performed by Muḥyī al-Dīn al-Maghribī at the Maragha observatory between 1262 and 1274 AD. In a treatise entitled Talkhīṣ al-majisṭī, preserved in a unique copy at Leiden, Universiteitsbibliotheek, Muḥyī al-Dīn explains his observations and measurements of the Sun, the Moon, the superior planets, and eight reference stars. His measurements of the meridian altitudes of the Sun, the superior planets, and the eight bright stars were made using the mural quadrant (...) of the observatory, and the times of their meridian transit using a water clock. The mean absolute error in the meridian altitudes of the Sun is ~ 3.1′, of the superior planets ~ 4.6′, and of the eight fixed stars ~ 6.2′. The clepsydras used by Muḥyī al-Dīn could apparently fix time intervals with a precision of ± 5 min. His estimation of the magnitudes of three lunar eclipses observed in Maragha in 1262, 1270, and 1274 AD is in close agreement with modern data. (shrink)

We devise imperative programming languages for verified real number computation where real numbers are provided as abstract data types such that the users of the languages can express real number computation by considering real numbers as abstract mathematical entities. Unlike other common approaches toward real number computation, based on an algebraic model that lacks implementability or transcendental computation, or finite-precision approximation such as using double precision computation that lacks a formal foundation, our languages are devised based on computable analysis, (...) a foundation of rigorous computation over continuous data. Consequently, the users of the language can easily program real number computation and reason about the behaviours of their programs, relying on their mathematical knowledge of real numbers without worrying about artificial roundoff errors. As the languages are imperative, we adopt precondition–postcondition-style program specification and Hoare-style program verification methodologies. Consequently, the users of the language can easily program a computation over real numbers, specify the expected behaviour of the program, including termination, and prove the correctness of the specification. Furthermore, we suggest extending the languages with other interesting continuous data, such as matrices, continuous real functions, et cetera.Abstract taken directly from the thesis.E-mail: [email protected]: https://sewonpark.com/thesis. (shrink)

While practical efforts in the field of artificial intelligence grow exponentially, the truly scientific and mathematically exact understanding of the underlying phenomena of intelligence and consciousness is still missing in the conventional science framework. The inevitably dominating empirical, trial-and-error approach has vanishing efficiency for those extremely complicated phenomena, ending up in fundamentally limited imitations of intelligent behaviour. We provide the first-principle analysis of unreduced many-body interaction process in the brain revealing its qualitatively new features, which give rise to (...) rigorously defined chaotic, noncomputable, intelligent and conscious behaviour. Based on the obtained universal concepts of unreduced dynamic complexity, intelligence and consciousness, we derive the universal laws of intelligence applicable to any kind of intelligent system interacting with the environment. We finally show why and how these fundamentally substantiated and therefore practically efficient laws of intelligent system dynamics are indispensable for correct AI design and training, which is urgently needed in this time of critical global change towards the truly sustainable development. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Pain in the Patient’s KneeMary Jacobus* (bio)We know very little about pain either.—Sigmund Freud, Inhibitions, Symptoms, and AnxietyPain cannot be absent from the personality.—Wilfred Bion, The Elements of Psycho-AnalysisBetween Therapy and HermeneuticsWhat is the place of a psychoanalysis that exists “between” therapy (considered both as a theory and a practice, but also as a theory of practice) and hermeneutics, or the theory of interpretation and understanding? How do (...) we understand psychoanalytic “understanding” itself, as a mental process that involves both the analyst and the analysand in the analytic encounter? I want to approach these related questions by way of the writing of the British psychoanalyst Wilfred Bion (1897–1979). Bion is best known outside the psychoanalytic profession as a pioneer of the so-called leaderless group and a theorist of group process [see Bion, Experiences in Groups]. But his collected theoretical and clinical oeuvre represents—especially in its “epistemological” period during the early 1960s, from Learning from Experience (1962) to Transformations (1965)—the most systematic, original, and philosophical revision of psychoanalytic theory since Melanie Klein’s controversial development of Freudian ideas during the middle decades of this century.1 Consequently, Bion’s influence pervades, and in some cases underpins, the rethinking of psychoanalytic theory and practice that characterizes British object relations in the wake of Klein. In particular, it gives contemporary post-Kleinian analysis its distinctive focus on (not) learning, (non)meaning, and (un)knowing, with an accompanying emphasis on the destruction of the links that make thought possible, as well as on the importance of being able to learn from experience and being able to bear not knowing. In other words, it brings to the fore difficulties associated specifically with questions involving interpretation and understanding in an analytic context.Bion’s writing is noted for its paradoxical combination of directness and difficulty, and for its emphasis on primitive emotional states and experiences, on the one hand, and epistemology and consciousness, on the other. Rigorous in its clinical focus, it also makes [End Page 99] frequent reference to philosophy, mathematics, and other branches of contemporary science. André Green, in the context of the new impetus that Bion gave to orthodox Kleinianism by linking it back to Freud, suggests that his particular contribution lay in realizing that the unconscious phantasies and archaic anxieties emphasized by Klein coincided with thought processes. Finding in Bion “an author who could measure up to Lacan,” Green saw the two as engaged in the common project of “reformulating psychoanalytic theory within a contemporary epistemological framework.” But for Green, “Lacan made the error of excessive abstraction, whereas Bion, with his constant reference to affect and the process of transformation, is the more authentically intellectual” [10–11].2 Although Bion’s writing is read wherever the theory and practice of British object relations psychoanalysis is seriously taught and studied, for the most part it remains surprisingly unknown within the academy. This is especially the case in America—even when it comes to literary critics and theorists with an interest in the “authentically intellectual,” who are often, by contrast, steeped in Lacan and well versed in recent Continental psychoanalytic writing. If only for this reason, but also because Bion’s work is particularly relevant to issues involving the relation between therapy and hermeneutics, I want to focus on the culminating book of Bion’s “epistemological” period during the early 1960s, Transformations—so called because of Bion’s analogy between the psychoanalyst’s representation of the experience of the analytic encounter and the artist’s transformation of the experience of what he sees.Bion’s writing, although terse and at times humorous, makes considerable demands on the reader, particularly on the capacity of the mind to entertain a new thought or consider it from a radically new perspective. He places pressure on thinking (including his own), as well as attempting to communicate the seeming incomprehensibility or hallucinatory dimensions of psychotic thought processes (which are often continuous with normal thought processes). In what follows, I will address both the kind of communication made by Bion himself—which is subject to its own interpretive difficulties and obscurities—and the relation between therapy... (shrink)

Several scholars, including Martin Hengel, R. Alan Culpepper, and Richard Bauckham, have argued that Papias had knowledge of the Gospel of John on the grounds that Papias’s prologue lists six of Jesus’s disciples in the same order that they are named in the Gospel of John: Andrew, Peter, Philip, Thomas, James, and John. In “A Note on Papias’s Knowledge of the Fourth Gospel” (JBL 129 [2010]: 793–794), Jake H. O’Connell presents a statistical analysis of this argument, according to which (...) the probability of this correspondence occurring by chance is lower than 1%. O’Connell concludes that it is more than 99% probable that this correspondence is the result of Papias copying John, rather than chance. I show that O’Connell’s analysis contains multiple mistakes, both substantive and mathematical: it ignores relevant evidence, overstates the correspondence between John and Papias, wrongly assumes that if Papias did not know John he ordered the disciples randomly, and conflates the probability of A given B with the probability of B given A. In discussing these errors, I aim to inform both Johannine scholarship and the use of probabilistic methods in historical reasoning. (shrink)

In this essay, I explore Alain Badiou’s longstanding project of theorizing political situations and political transformation through the analysis of forms and formalisms. This amounts, I argue, to a politics of form that draws on the thought of Sartre, Althusser, and Lacan, but offers new alternatives for political thought and action today. In particular, Badiou’s rigorous consideration of forms, which draws on mathematics, model theory, set theory, and category theory, allows him to theorize political change in a way (...) that avoids the errors of existing critical approaches. Central to this consideration are Badiou’s theory of what he calls the event, which summons itself into existence by its own naming of itself or auto‐nomination, and his recent theory of the plurality of worlds and their possible transformation in Logics of Worlds. I also discuss Badiou’s recent treatment of the 20th century project of formalization in The Century, arguing that Badiou’s analysis here offers useful terms for the political thought of the present situation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Strange Death of British IdealismEdward SkidelskyIIn 1958, the Oxford philosopher G. J. Warnock opened his survey of twentieth-century English philosophy with some disparaging comments on British Idealism. It was, he writes, "an exotic in the English scene, the product of a quite recent revolution in ways of thought due primarily to German influences." Analytic philosophy, by contrast, represents a return to the venerable lineage of British empiricism, as (...) Bertrand Russell "symbolically joins hands" with his godfather John Stuart Mill. "Hume and Berkley would have been sadly puzzled by the pages of Bradley, to say nothing of Hegel's. But either might have conversed quite naturally with Moore, and with Russell too, at least in his less technical moments."1All this is rhetoric rather than history. Russell can be called the inheritor of Mill only with respect to his political and social attitudes; his main philosophical endeavour—the establishment of mathematics on a priori foundations—ran diametrically opposed to Mill. And Hume and Berkley would have been just as sadly puzzled by Moore's analysis of language—an activity they both regarded as the defining vice of scholasticism—as by anything in Bradley.Warnock's error has proved remarkably persistent. Only recently, thanks largely to the work of Peter Hylton, has the anti-empiricist, Platonist character of Russell's early thought been properly appreciated.2 I do not intend, in what follows, to retrace these philosophical arguments. My question is historical. It concerns the role of national feeling, first in the demise of Idealism after the First World War, and later in the interpretation of that demise after the Second. With regard to first question, I argue that the rejection of Idealism in Britain was in no sense [End Page 41] a reassertion of national tradition. The opposite is closer to the truth; the change in philosophical fashion was primarily a consequence of a widespread revulsion against nationalism following World War I. That this was later misinterpreted owes largely—and this is my second claim—to the very different climate prevailing in Britain after World War II. If in the twenties the British reserved their scorn for "Prussian militarism," in the fifties everything associated with Germany, its philosophy included, was viewed as irredeemably tainted. Analytic philosophy tried to present itself as a home-grown product, with deep roots in the English empiricist tradition. This partly accounts for the change in style from the highly technical scientism of Russell and the logical positivists to the concern with "ordinary language" characteristic of post-war Oxford philosophy. The fact that most of analytic philosophy's continental exponents were forced by Hitler into British or American exile assisted in the process of its naturalisation. Within a decade, it was possible to regard "analytic" and "Anglo-Saxon" philosophy as synonyms.Warnock's mistake now becomes easier to understand. He has interpreted the early twentieth century in the light of the post-war period, coming to the conclusion that Moore and Russell's rejection of Idealism was a reassertion of English national tradition. He has condensed fifty years of history into a single moment. I intend in what follows to pull apart the stages that Warnock has concertinaed together, demonstrating that national feeling played a far more complex and contradictory role in the history of British philosophy.IIOn one point Warnock is correct: Idealism did not perish of refutation. "Metaphysical systems do not yield...," he writes, "to frontal attack. Their odd property of being demonstrable only, so to speak, from within confers on them also a high resistance to attack from outside.... Such systems are much more vulnerable to ennui than to disproof" (Warnock, pp. 10–11). The claim of Russell and Moore to have "refuted" Bradley is regarded with scepticism by most recent commentators. Idealism, they point out, rested on something more substantial than a mere failure of logic or common-sense, and was therefore not vulnerable to the kind of simple knock-down arguments advanced by Russell and Moore. The causes of its decline must be sought elsewhere, in the broader cultural and political tendencies of the age.Had Russell and Moore's arguments been truly knock-down... (shrink)

In coal mining industry, the running state of mine ventilators plays an extremely significant role for the safe and reliable operation of various industrial productions. To guarantee the better reliability, safety, and economy of mine ventilators, in view of early detection and effective fault diagnosis of mechanical faults which could prevent unscheduled downtime and minimize maintenance fees, it is imperative to construct some viable mathematical models for mine ventilator fault diagnosis. In this article, we plan to establish a data-based mine (...) ventilator fault diagnosis method to handle situations where engineers are absent or they are incapable of coming to a conclusion from multisource data. In the process of building the mine ventilator fault diagnosis model, considering that probabilistic rough sets could reduce the errors triggered by incompleteness, inconsistency, and inaccuracy without needing any additional assumptions and Pythagorean fuzzy multigranulation rough sets over the two universes’ model could effectively handle data representation, fusion, and analysis issues, we generalize the existing PF MGRSs over the two universes’ model to the PRS setting, as well as to further establish a novel model named Pythagorean fuzzy multigranulation probabilistic rough sets over two universes. In the granular computing paradigm, three types of PF MG-PRSs over two universes based on the risk attitude of engineers are proposed at first. Afterwards, several basic propositions of the newly proposed model are explored. Moreover, a PF multigranulation probabilistic model for mine ventilator fault diagnosis based on PF MG-PRSs over two universes is investigated. At last, a real-world case study of dealing with a mine ventilator fault diagnosis problem is given to illustrate the practicality of the presented model, and a validity test, a sensitivity analysis, and a comparison analysis are further explored to demonstrate the effectiveness of the presented model. (shrink)

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show that (...) by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Cambridge Companion to Bertrand RussellPeter H. DentonNicholas Griffin, editor. The Cambridge Companion to Bertrand Russell. New York: Cambridge University Press, 2003. Pp. xvii + 550. Cloth, $75.00. Paper, $26.00.It is a daunting task to conceive of a single companion to Bertrand Russell, who in life as in thought was never content with a single anything. Nicholas Griffin has brought his customary expertise to the project, and in (...) snaring fourteen other contributors to this volume, has produced a welcome addition to an academic's library of Russell materials.The shape of the volume, its contents and its presentation, however, unfortunately reflect the lopsided nature of current Russell studies. One hopes for a companion volume that has the breadth of the earlier volume edited fifty years ago by Paul Schilpp, but in vain. This book is overwhelmingly about Russell before 1918, almost entirely about philosophy of mathematics, epistemology and logic, and several of the pieces easily could have been included in a festschrift for Frege instead.What is included is well done, but to call it "the most convenient and accessible guide to Russell available" is to speak in comparative terms. To be fair, it sets out to focus on a narrow definition of Russell's contributions to modern philosophy, and the contributors accomplish this task. Ian Grattan-Guinness writes on "Mathematics in and behind Russell's Logicism, and its Reception," a subject that continues into "Bertrand Russell's Logicism" by Martin Godwyn and Andrew D. Irvine. The roots of analytic philosophy are of primary concern in this volume, so Richard L. Cartwright explores "Russell and Moore, 1898-1905" and Michael Beaney relates "Russell and Frege." "The Theory of Descriptions" by Peter Hylton sets up two pieces on "Russell's Substitutional Theory" (Gregory Landini) and "The Theory of Types" (Alasdair Urquhart), while Paul Hager's "Russell's Method of Analysis" leads into "Russell's Neutral Monism" (R.E.Tully) and "The Metaphysics of Logical Atomism" (Bernard Linsky).The next portion of the book expands the intellectual focus somewhat, with "Russell's Structuralism and the Absolute Description of the World" by William Demopoulos—perhaps [End Page 349] the best in a series of good papers—and "From Knowledge by Acquaintance to Knowledge by Causation" by Thomas Baldwin. Approaching the end of the book, Anthony Grayling's essay on "Russell, Experience, and the Roots of Science" remains firmly planted in the pre-1918 milieu, though his tentative commentary on Human Knowledge at least glimmers the extension of Russell's intellectual life into the post-Great War period.Effectively the entire second half of Russell's life is left to the final contributor, Charles Pidgen, for commentary. Author of Russell on Ethics, much of Pigden's short essay here ("Bertrand Russell: Moral Philosopher or Unphilosophical Moralist?") somewhat plaintively directs the reader to his book for an elaboration of why Russell is worth reading after 1918, at least on the subject of ethics. Of the other authors, only Hager (317) observes the extent to which Russell's post-War work mistakenly has been ignored.The book concludes with a selective bibliography. The audience, for the most part, is a specialist one, with only some essays accessible to the general reader. On that score, Griffin's introduction and his own essay, "Russell's Philosophical Background," are easily the most readable. At the risk of quibbling, I was disturbed by the uneven production of the book, as each author set his own standard for footnotes, endnotes and bibliography (creating a jumble of styles). Even within individual contributions, there are a number of inconsistencies in notation as well as typographical errors.By the end, I was left hoping there might be a second companion volume for Bertrand Russell, one that reflects what he spent the last fifty years of his life thinking, writing, and doing. In the absence of such material (which is of far greater popular interest today just as it was in Russell's lifetime), this book by itself is more of a companion to current Russell studies than to Bertrand Russell himself.Peter H. DentonThe University of WinnipegCopyright © 2004 Journal of the History of Philosophy, Inc.... (shrink)