This paper provides a philosophical analysis of the ongoing controversy surrounding R.A. Fisher's famous ‘fundamental theorem’ of natural selection. The difference between the ‘traditional’ and ‘modern’ interpretations of the theorem is explained. I argue that proponents of the modern interpretation have captured Fisher's intended meaning correctly and shown that the theorem is mathematically correct, pace the traditional consensus. However, whether the theorem has any real biological significance remains an unresolved issue. I argue that (...) the answer depends on whether we accept Fisher's non-standard notion of environmental change, on which the theorem rests; arguments for and against this notion are explored. I suggest that there is a close link between Fisher's fundamental theorem and the modern ‘gene's eye’ view of evolution. Introduction What Does the Fundamental Theorem Say? Key Concepts Explained Alleged Significance of the FTNS Traditional versus Modern Interpretations of the FTNS The Modern Interpretation Illustrated Fisher's Concept of ‘Environmental Change’ Causality and the Modern Interpretation The Significance of the FTNS Re-considered Appendix CiteULike Connotea Del.icio.us What's this? (shrink)

Fisher’s ‘fundamental theorem of natural selection’ is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher’s general project for biology, and analyze why it was so very fundamental for Fisher. I defend Ewens and Lessard in the view that the theorem is in fact a true theorem if, as Fisher claimed, ‘the terms employed’ are ‘used strictly (...) as defined’. Finally, I explain the role that projects such as Fisher’s play in the progress of scientific inquiry. (shrink)

Fisher’s ‘fundamental theorem of natural selection’ is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher’s general project for biology, and analyze why it was so very fundamental for Fisher. I defend Ewens and Lessard in the view that the theorem is in fact a true theorem if, as Fisher claimed, ‘the terms employed’ are ‘used strictly (...) as defined’. Finally, I explain the role that projects such as Fisher’s play in the progress of scientific inquiry. (shrink)

Fisher’s ‘fundamental theorem of natural selection’ is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher’s general project for biology, and analyze why it was so very fundamental for Fisher. I defend Ewens (1989) and Lessard (1997) in the view that the theorem is in fact a true theorem if, as Fisher claimed, ‘the terms employed’ are (...) ‘used strictly as defined’ (1930, p. 38). Finally, I explain the role that projects such as Fisher’s play in the progress of scientific inquiry. (shrink)

The idea that natural selection can operate on cultural as well as genetic variation is central to recent theories of cultural evolution. This raises an overarching question: how much of traditional evolutionary theory, which was formulated in population-genetic terms, can survive intact once the possibility of cultural inheritance is taken into account? This question is addressed in relation to R. A. Fisher’s “fundamental theorem” of natural selection. Though Fisher’s theorem may appear to be an essentially genetic result, (...) a version of the theorem can be derived that applies to any evolving population, with an arbitrary inheritance mechanism. However, I argue that the biological significance of this “generalized” version of the fundamental theorem is less than that of the Fisherian original. (shrink)

Freedom and the Rule of Law takes a critical look at the historical beginnings of law in the United States, and how that history has influenced current trends regarding law and freedom. Anthony Peacock has compiled articles that examine the relationship between freedom and the rule of law in America. The rule of law is fundamental to all liberal constitutional regimes whose political orders recognize the equal natural rights of all.

The relationship between Fisher's fundamental theorem of natural selection and the ecological environment of density regulation is examined. Using a linear model, it is shown that the theorem holds when density regulation is caused by exploitative competition and that the theorem fails with interference competition. In the latter case the theorem holds only at the limit of zero population density and/or at the limit where the competitively superior individuals cannot monopolise the resource. The results (...) are discussed in relation to population dynamics and life history evolution, where evidence suggests that the level of interference competition in natural populations is so high that the fundamental theorem does not apply. (shrink)

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that (...) this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory. (shrink)

Revised to reflect the current status of scientific and professional theory, practices, and debate across all facets of ethical decision making, this latest edition of Celia B. Fisher's acclaimed book demystifies the American Psychological Association's (APA) Ethical Principles of Psychologists and Code of Conduct. The Fifth Edition explains and puts into practical perspective the format, choice of wording, aspirational principles, and enforceability of the code. Providing in-depth discussions of the foundation and application of each ethical standard to the broad (...) spectrum of scientific, teaching, and professional roles of psychologists, this unique guide helps practitioners effectively use ethical principles and standards to morally conduct their work activities, avoid ethical violations, and, most importantly, preserve and protect the fundamental rights and welfare of those whom they serve. This edition retains and expands upon the critical content of the previous editions to help readers apply the Ethics Code to contemporary social issues in the conduct of responsible psychological science and practice. (shrink)

I apply the notion of truthmaking to the topic of fundamentality by articulating a truthmaker theory of fundamentality according to which some truths are truth-grounded in certain entities while the ones that don't stand in a metaphysical-semantic relation to the truths that do. I motivate this view by critically discussing two problems with Ross Cameron's truthmaker theory of fundamentality. I then defend this view against Theodore Sider's objection that the truthmaking approach to fundamentality violates the purity constraint. Truthmaker theorists can (...) have a trouble-free theory of fundamentality. (shrink)

The revival of analytic metaphysics in the latter half of the twentieth century is typically understood as a consequence of the critiques of logical positivism, Quine’s naturalization of ontology, Kripke’s Naming and Necessity, clarifications of modal notions in logic, and the theoretical exploitation of possible worlds. However, this explanation overlooks the work of metaphysicians at the height of positivism and linguisticism that affected metaphysics of the late twentieth century. Donald C. Williams is one such philosopher. In this paper I explain (...) how Williams’s fundamental ontology and philosophy of time influenced in part the early formation of David Lewis’s metaphysics. Thus, Williams played an important role in the revival of analytic metaphysics. (shrink)

In this book, Naomi Fisher provides a cohesive interpretation of Schelling's philosophical work from 1792-1802 as a mystical Platonism. According to this interpretation, Schelling is guided by two overarching commitments during this time. First, Schelling is committed to mysticism regarding the absolute. That is, the absolute is ineffable; it cannot be described in conceptual terms. For this reason, it remains inferentially external to any given philosophical system. Second, Schelling is committed to a priority monism: All things are grounded in the (...) absolute, but finite things possess an integral unity all their own, and so have a distinct and relatively independent existence. This book shows how these two commitments cohere in a project that fulfils the distinctive aims of post-Kantian philosophy, and it traces their inspiration to Schelling's early engagement with Plato's dialogues and to his education, which was Neoplatonic in orientation. In presenting Schelling's philosophy of this decade as guided by these two fundamental commitments, this book poses a challenge to those readings of Schelling's philosophy according to which it shifts frequently in its basic commitments during this time. According to the interpretation presented in this book, Schelling's appropriations of various strands of Platonism distinguish him from his contemporaries and give rise to his idiosyncratic approach to the projects of post-Kantian philosophy. Mysticism, Platonism, monism, Schelling, Neoplatonism, philosophy of nature, identity philosophy, German Idealism, Romanticism In this book, Naomi Fisher provides a cohesive interpretation of Schelling's philosophical work from 1792-1802 as a mystical Platonism. According to this interpretation, Schelling is guided by two overarching commitments during this time. First, Schelling is committed to mysticism regarding the absolute. That is, the absolute is ineffable; it cannot be described in conceptual terms. For this reason, it remains inferentially external to any given philosophical system. Second, Schelling is committed to a priority monism: All things are grounded in the absolute, but finite things possess an integral unity all their own, and so have a distinct and relatively independent existence. This book shows how these two commitments cohere in a project that fulfils the distinctive aims of post-Kantian philosophy, and it traces their inspiration to Schelling's early engagement with Plato's dialogues and to his education, which was Neoplatonic in orientation. In presenting Schelling's philosophy of this decade as guided by these two fundamental commitments, this book poses a challenge to those readings of Schelling's philosophy according to which it shifts frequently in its basic commitments during this time. According to the interpretation presented in this book, Schelling's appropriations of various strands of Platonism distinguish him from his contemporaries and give rise to his idiosyncratic approach to the projects of post-Kantian philosophy. Mysticism, Platonism, monism, Schelling, Neoplatonism, philosophy of nature, identity philosophy, German Idealism, Romanticism In this book, Naomi Fisher provides a cohesive interpretation of Schelling's philosophical work from 1792-1802 as a mystical Platonism. According to this interpretation, Schelling is guided by two overarching commitments during this time. First, Schelling is committed to mysticism regarding the absolute. That is, the absolute is ineffable; it cannot be described in conceptual terms. For this reason, it remains inferentially external to any given philosophical system. Second, Schelling is committed to a priority monism: All things are grounded in the absolute, but finite things possess an integral unity all their own, and so have a distinct and relatively independent existence. This book shows how these two commitments cohere in a project that fulfils the distinctive aims of post-Kantian philosophy, and it traces their inspiration to Schelling's early engagement with Plato's dialogues and to his education, which was Neoplatonic in orientation. In presenting Schelling's philosophy of this decade as guided by these two fundamental commitments, this book poses a challenge to those readings of Schelling's philosophy according to which it shifts frequently in its basic commitments during this time. According to the interpretation presented in this book, Schelling's appropriations of various strands of Platonism distinguish him from his contemporaries and give rise to his idiosyncratic approach to the projects of post-Kantian philosophy. Mysticism, Platonism, monism, Schelling, Neoplatonism, philosophy of nature, identity philosophy, German Idealism, Romanticism. (shrink)

This essay describes a performance by the Greek theatre collective, Blitz Theatre – Late Night – as constituting a theatrical response to current political crises in Europe. What I call a ‘theatre of the impasse’ seeks to bear witness to the experience of impasse, where impasse and crisis must be fundamentally distinguished. Impasse is revealed where crisis admits of no decision adequate to the situation; and, correspondingly, where theatre loses faith in the power of decision to resolve its conflicts. I (...) situate these claims with reference to Carl Schmitt’s and Walter Benjamin’s dispute over political theology, arguing that a theatre of the impasse might be thought as an ‘allegorical’ theatre in Benjamin’s terms. Blitz Theatre’s Late Night reveals, thereby, the concealed truth of the impasse: a founding human sociality experienced as world immanence. In doing so doing, I argue, this theatre frustrates every hope for the kind of political theology of the stage envisaged by Schmitt. I read the performance, instead, as an elegy to Nancy’s inoperative community, at the centre of which are the figure of lovers, bound to, yet unable to take possession of, one another. Staging impasse, Late Night allegorises the fragile human community, exposed in its fundamental precarity. (shrink)

A collection of transcripts from Mark Fisher's final series of lectures at Goldsmiths, University of London, in late 2016. Edited with an introduction by Matt Colquhoun, this collection of lecture notes and transcriptions reveals acclaimed writer and blogger Mark Fisher in his element -- the classroom -- outlining a project that Fisher's death left so bittersweetly unfinished. Beginning with that most fundamental of questions -- "Do we really want what we say we want?" -- Fisher explores the (...) relationship between desire and capitalism, and wonders what new forms of desire we might still excavate from the past, present, and future. From the emergence and failure of the counterculture in the 1970s to the continued development of his left-accelerationist line of thinking, this volume charts a tragically interrupted course for thinking about the raising of a new kind of consciousness, and the cultural and political implications of doing so. For Fisher, this process of consciousness raising was always, fundamentally, psychedelic -- just not in the way that we might think... (shrink)

Performance philosophy commences with an impertinent gesture when it describes itself as inaugurating a ‘new field’ of study. Accompanying that claim is a radical proposition that ‘performance thinks’; that it should be counted as a form of philosophising in its own right. But in what sense can performance be construed as ‘genuinely’ philosophical thought? Taking my cue from Laura Cull’s alignment of performance philosophy with Laruelle’s practice of ‘non philosophy’ – and specifically, with its introduction of ‘democracy’ into the dispositives (...) of ‘standard’ philosophy in order to challenge its transcendental authority over the Real – I argue that performance philosophy might be seen to enact a similar disruption of the ‘dispositives’ of performance theory. This, however, is only partly what is at stake in the fundamental proposition of performance philosophy, and I conclude by suggesting that a more radical proposal lies behind its assertion of a new ‘field’ – one that does not reduce it to an empirical fact, but grasps it as a radical ‘utopian’ hypothesis designed to ‘open up’ the philosophical dimension of performance itself; utopian because what performance offers – seen in this way - is not simply another system of representation but a possible democratic thought of the Real itself. (shrink)

Current framing of the debate over the privatization of the State’s legislative and adjudicative functions masks the fact that there are distinct and conflicting versions of privatization of law. The different privatization models diverge on fundamental questions relating to the ontology of law, the role of social cooperation mechanisms in the lives of people, as well as the types of private legislative and adjudicative institutions that ought to replace the State’s legal system. In light of such conflicting normative premises, (...) the distinct models of the privatization of law pose different kinds of challenges to both proponents and opponents of the privatization of law. At the outset, the Article juxtaposes two distinct visions regarding the privatization of law and adjudication—the market-based privatization model versus the community-based model. This analytical framework is then used to offer a fresh look at the privatization of law debate. The Article shows that the distinctions between the privatization models, especially with respect to the depiction of the social agents that are to replace the state’s lawmaking capacity, have great bearing on the advantages and disadvantages associated with privatization of law, and generate different types of costs and benefits. More specifically, the Article shows that the market-based model has greater susceptibility to market failures and to the under-provision of the public goods associated with the enterprise of law than the community-based paradigm. In addition, the market-based model runs a higher risk of corrupting the prevailing understanding of law as a collective, meaning-creating enterprise. The community-based model, on the other hand, has a greater vulnerability for coercion and is also more prone to political failures and public choice problems. (shrink)

This chapter offers an analysis of the speech conditions constitutive for the staging of political disagreement. Rather than seeking to offer an explanation for various situations of protest, however, it aims to identify what, if anything, is unique or peculiar to such modes of address. Drawing on the resources of speech act theory, the chapter suggests a reading of ‘dissensual speech’ as a form of ‘unauthorised’ speech through which the ‘people’ appear, however, evanescently. It analyses the peculiarities of dissensual speech (...) in the following ways: first, in terms of specifying its modes of utterance, which are identified with phatic and agonic modes of address and, second, in terms of its performative attitudes. In analysing the latter, I turn to Foucault’s notion of parrēsiastic speech in order to confront a paradox that arises with my reading of dissensual speech viewed in terms of ‘performatives’, since parrēsia—‘speaking truth to power’—is radically opposed to two fundamental rules that govern performatives and illocutionary forces: that they are conventional and that their enunciator must be authorised to use them. The chapter proposes a resolution to this contradiction by showing that an affinity exists between parrēsiastic speech and dissensual speech insofar as both entail ‘risk’ to the speaker in uttering a truth. The question then is whether it is possible to collectivise that risk, or whether parrēsia necessarily remains the speech of the individual martyr. (shrink)

This chapter offers an analysis of the speech conditions constitutive for the staging of political disagreement. Rather than seeking to offer an explanation for various situations of protest, however, it aims to identify what, if anything, is unique or peculiar to such modes of address. Drawing on the resources of speech act theory, the chapter suggests a reading of ‘dissensual speech’ as a form of ‘unauthorised’ speech through which the ‘people’ appear, however, evanescently. It analyses the peculiarities of dissensual speech (...) in the following ways: first, in terms of specifying its modes of utterance, which are identified with phatic and agonic modes of address and, second, in terms of its performative attitudes. In analysing the latter, I turn to Foucault’s notion of parrēsiastic speech in order to confront a paradox that arises with my reading of dissensual speech viewed in terms of ‘performatives’, since parrēsia—‘speaking truth to power’—is radically opposed to two fundamental rules that govern performatives and illocutionary forces: that they are conventional and that their enunciator must be authorised to use them. The chapter proposes a resolution to this contradiction by showing that an affinity exists between parrēsiastic speech and dissensual speech insofar as both entail ‘risk’ to the speaker in uttering a truth. The question then is whether it is possible to collectivise that risk, or whether parrēsia necessarily remains the speech of the individual martyr. (shrink)

Performance philosophy commences with an impertinent gesture when it describes itself as inaugurating a ‘new field’ of study. Accompanying that claim is a radical proposition that ‘performance thinks’; that it should be counted as a form of philosophising in its own right. But in what sense can performance be construed as ‘genuinely’ philosophical thought? Taking my cue from Laura Cull’s alignment of performance philosophy with Laruelle’s practice of ‘non philosophy’ – and specifically, with its introduction of ‘democracy’ into the dispositives (...) of ‘standard’ philosophy in order to challenge its transcendental authority over the Real – I argue that performance philosophy might be seen to enact a similar disruption of the ‘dispositives’ of performance theory. This, however, is only partly what is at stake in the fundamental proposition of performance philosophy, and I conclude by suggesting that a more radical proposal lies behind its assertion of a new ‘field’ – one that does not reduce it to an empirical fact, but grasps it as a radical ‘utopian’ hypothesis designed to ‘open up’ the philosophical dimension of performance itself; utopian because what performance offers – seen in this way - is not simply another system of representation but a possible democratic thought of the Real itself. (shrink)

In Summa Theologiae I.76.1 Aquinas presents an argument for the hylomorphic union of body and soul that he attributes to Aristotle. Aquinas builds on Aristotle’s original argument, however, offering his own short but powerful line of reasoning in support of one of the main premises. This additional argument involves an appeal to the principle that nothing acts except insofar as it is in act. This principle has roots in the thought of Aristotle, but is not explicitly used by him. It (...) is, however, fundamental for Aquinas and pervasive throughout his work. In this paper I examine the principle and its implications for Aquinas’ version of the argument. Furthermore, I argue that the principle is foundational to Aquinas’ criticisms of Averroes’ account of the intellective soul and that its inclusion renders Aquinas’ version of the argument incompatible with Averroes’ view. (shrink)

Explores a postmodern criticism of P. S. Churchland's claims regarding materialism. Materialism is classically understood to be the philosophical position which holds that matter is the fundamental reality of the world, and so neurobiological explanations can be said to be materialistic. Neurobiological explanations of behavior are used increasingly in the place of psychological explanations. This trend is indicative of the rise in popularity of materialism. Churchland is one of the intellectual leaders in the modern manifestation of materialism. She is (...) a self-proclaimed materialist and "neurophilosopher." In order for Churchland to maintain her materialist position, she must assume a transparency in scientific method, an assumption which is a legacy of Enlightenment philosophy. However, many postmodern philosophers including M. Heidegger and H. G. Gadamer have questioned this assumed transparency of method. The implications of this postmodern philosophy for the current materialistic trends in psychology are discussed. 2012 APA, all rights reserved). (shrink)

A recurring theme dominates recent philosophical debates about the nature of conscious perception: naïve realism’s opponents claim that the view is directly contradicted by empirical science. I argue that, despite their current popularity, empirical arguments against naïve realism are fundamentally flawed. The non-empirical premises needed to get from empirical scientific findings to substantive philosophical conclusions are ones the naïve realist is known to reject. Even granting the contentious premises, the empirical findings do not undermine the theory, given its overall philosophical (...) commitments. Thus, contemporary empirical research fails to supply any new argumentative force against naïve realism. I conclude that, as philosophers of mind, we would be better served spending a bit less time trying to wield empirical science as a cudgel against our opponents, and a bit more time working through the implications of each other’s views – something we can accomplish perfectly well from the comfort of our armchairs. (shrink)

The Networks of Centres of Excellence programme is perhaps Canada's most dramatic science policy innovation since theFirst World War. This article traces its development, using documentary analysis and interviews with the policy actors responsible for conceiving and implementing the programme.Established in 1989, the networks were explicitly designed to change the norms of science. The intention was to instil an approach to long-term fundamental research that considered possibilities of use from the start. Of equal importance was the idea that management (...) was essential to produce knowledgein a network structure. (shrink)

Using Heijenoort’s unpublished generalized rules of quantification, we discuss the proof of Herbrand’s Fundamental Theorem in the form of Heijenoort’s correction of Herbrand’s “False Lemma” and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand’s Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it “the central theorem of predicate logic” and considered the (...) form of its expression to be “concise and felicitous”. (shrink)

In a fluent, clear, and lively style this translation by two of Maslov's junior colleagues brings the work of the late Soviet scientist S. Yu. Maslov to a wider audience. Maslov was considered by his peers to be a man of genius who was making fundamental contributions in the fields of automatic theorem proving and computational logic. He published little, and those few papers were regarded as notoriously difficult. This book, however, was written for a broad audience of (...) readers and describes elegant examples of applications in such fields as computer science, artificial intelligence, operations research, economic modeling, and biological modeling, among others. The book also brings to light the work by the American mathematician E. L. Post, which inspired Maslov's own work in the development of a general theory and which has been long neglected by mathematicial logicians and systems theorists in the United States. The book's first chapter introduces the Rules of the Game. Part I, Mathematics of Calculi, covers E. L. Post's canonical systems, deductive systems and algorithms, and probabilistic calculi and deductive information. Part II, Horizonal Modeling, takes up a "toy" economy, the calculi of technological possibilities, and the development of rules. Part III, Vertical Modeling, deals with the topics of "to fight and to search" and the consequences of the asymmetry of cognitive mechanisms. Vladimir Lifschitz is affiliated with the Department of Computer Science at Stanford University, and Michael Gelfond with the Department of Electrical Engineering and Computer Science at the University of Texas, El Paso. Theory of Deductive Systems and Its Applicationsis included in the Foundation of Computing Series, edited by Michael Garey. (shrink)

The notion that natural selection is a process of fitness maximization gets a bad press in population genetics, yet in other areas of biology the view that organisms behave as if attempting to maximize their fitness remains widespread. Here I critically appraise the prospects for reconciliation. I first distinguish four varieties of fitness maximization. I then examine two recent developments that may appear to vindicate at least one of these varieties. The first is the ‘new’ interpretation of Fisher's (...) class='Hi'>fundamental theorem of natural selection, on which the theorem is exactly true for any evolving population that satisfies some minimal assumptions. The second is the Formal Darwinism project, which forges links between gene frequency change and optimal strategy choice. In both cases, I argue that the results fail to establish a biologically significant maximization principle. I conclude that it may be a mistake to look for universal maximization principles justified by theory alone. A more promising approach may be to find maximization principles that apply conditionally and to show that the conditions were satisfied in the evolution of particular traits. (shrink)

It is demonstrated that a non Abelian Stokes Theorem is necessary to describe the B3 field of radiation. A simple form of the theorem is build up from the fundamental definition of B3 in O(3) gauge field theory, which is a gauge field theory applied to electrodynamics with an O(3) internal gauge symmetry bases on a complex basis ((1), (2), (3)). The indices (1) and (2) are complex conjugate pairs based on circular polarization, and the index (3) (...) is aligned with the propagation axis of radiation. (shrink)

The authors' novel approach to some interesting mathematical concepts - not normally taught in other courses - places them in a historical and philosophical setting. Although primarily intended for mathematics undergraduates, the book will also appeal to students in the sciences, humanities and education with a strong interest in this subject. The first part proceeds from about 1800 BC to 1800 AD, discussing, for example, the Renaissance method for solving cubic and quartic equations and providing rigorous elementary proof that certain (...) geometrical problems posed by the ancient Greeks cannot be solved by ruler and compass alone. The second part presents some fundamental topics of interest from the past two centuries, including proof of G del's incompleteness theorem, together with a discussion of its implications. (shrink)

In 1929 Carnap gave a paper in Prague on Investigations in General Axiomatics; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnap''s publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work (...) here,showing that it provides important insights intoCarnap''s development during this critical period, thetransition from Aufbau to Syntax,especially regarding the nature and motivation ofCarnap''s logicism. Moreover, we show how theAxiomatics influenced Carnap''s student Gödel inreaching the fundamental logical results that soonafterwards undermined Carnap''s project. (shrink)

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, (...) the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. (shrink)

In [This Zeitschrift 25 , 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order (...) Pavelka's logic is preserved under some direct product of lattices of truth values. (shrink)

The conformity of mathematics and physics has so far been taken for granted. Philosophical explanations of that fundamental fact have never been satisfactory, mathematical explanations never had been attempted. In the following a fundamental theorem for the conformity of mathematics and physics will be demonstrated.Mathematics can be defined as the science of Number, physics as the science of Matter. The elementary constituents of mathematics are the prime numbers, those of matter the particles, particularly protons and electrons. The (...) only essential property of these elements as such is their existence, given within a given complex or space. Their fundamental relation is their distribution within their respective spaces. The theorem of the distribution of primes in the arithmetical series of numbers is the fundamental theorem of mathematics. The fundamental theorem of physics would be a theorem of the distribution of particles in geometrical space. If it were shown that both theorems were equal, the essential conformity of mathematics and physics would be mathematically demonstrated. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

Legal scholarship exploring the nature of evidence and the process of juridical proof has had a complex relationship with formal modeling. As evident in so many fields of knowledge, algorithmic approaches to evidence have the theoretical potential to increase the accuracy of fact finding, a tremendously important goal of the legal system. The hope that knowledge could be formalized within the evidentiary realm generated a spate of articles attempting to put probability theory to this purpose. This literature was both insightful (...) and frustrating. Much light was shed on the legal system, but it also quickly became evident that the tools of probability theory were in many ways ill-constructed for the task. Fundamental incompatibilities between the structure of legal decision making and the extant formal tools were identified, and it became evident that many of the purported explanations of legal phenomena were internally inconsistent. As a consequence, interest in this type of formal modeling declined, and attention was directed toward different kinds of explanations of the phenomena. Perhaps under the influence of a recent trend toward various types of formal modeling in legal scholarship, a recent burst of articles, rather than attempting to explain the macro structure of trials, which was the previous object of interest, attempts to quantify the probative value of various items of evidence in ways consistent with the formal features of various probability theories, and then to study decision making from that perspective. For example, the value of evidence is often purported to be its likelihood ratio, that is, the probability of discovering or receiving the evidence given a hypothesis (e.g., the defendant did it) divided by the probability of discovering or receiving the evidence given the negation of the hypothesis (the defendant didn't do it). Alternatively, the value of evidence is purported (more contextually) to be the information gain it provides, defined as the increase in probability it provides for a hypothesis above the probability of the hypothesis based on the other available evidence. Both conceptions then assume that all of the various probability assessments conform or ought to conform to the dictates of Bayes' theorem (that maintains consistency among such assessments); empirical studies are then done testing the extent to which this is so and proposing how the law can increase the probability that it is so. The general criticisms of using Bayes' theorem as a formal model of juridical proof are well known and were integral to the last wave of interest in formal modeling of the evidentiary process. This paper thus for the most part puts aside that more general issue, and focuses specifically on mathematical modeling of the value of particular items of evidence. The paper demonstrates that formal modeling has only limited value in explaining the value of legal evidence, much more limited than those constructing and discussing the models assume, and thus that the conclusions they draw about the value of evidence are unwarranted. This is done through a discussion of four recent examples that attempt to quantify evidence relating to, respectively, carpet fibers, infidelity, DNA random-match evidence, and character evidence used to impeach a witness. This article thus makes two contributions. First, and most importantly, it is another demonstration of the complex relationship between algorithmic tools and legal decision making. Second, at a minimum it points out serious pitfalls for analytical or empirical studies of juridical proof. (shrink)

Quantum mechanics is seen to be incomplete not because it cannot explain the correlations that characterize entanglement without invoking either non-locality or realism, both of which, despite special relativity or no-go theorems, are at least conceivable. Quantum mechanics is incomplete, in a perhaps broader than hidden variable sense, because it fails to address within its theoretical structure the question of how even a single particle, by being in a given quantum state, causes the frequency distribution of measurement values specified by (...) the state. This incompleteness of quantum mechanics as it is currently conceived is both fundamental and indefeasible. Failure to address the question of how the states of entangled particles are given effect to yield the correlations they specify is simply a particular albeit attention arresting instance of this incompleteness. But if that is so then quantum mechanics cannot be held to be inconsistent with locality. (shrink)