Imagination occupies a central place in philosophy, going back to Aristotle. However, following a period of relative neglect there has been an explosion of interest in imagination in the past two decades as philosophers examine the role of imagination in debates about the mind and cognition, aesthetics and ethics, as well as epistemology, science and mathematics. This outstanding _Handbook_ contains over thirty specially commissioned chapters by leading philosophers organised into six clear sections examining the most important aspects of the philosophy (...) of imagination, including: Imagination in historical context: Aristotle, Descartes, Hume, Kant, Husserl, and Sartre What is imagination? The relation between imagination and mental imagery; imagination contrasted with perception, memory, and dreaming Imagination in aesthetics: imagination and our engagement with music, art, and fiction; the problems of fictional emotions and ‘imaginative resistance’ Imagination in philosophy of mind and cognitive science: imagination and creativity, the self, action, child development, and animal cognition Imagination in ethics and political philosophy, including the concept of 'moral imagination' and empathy Imagination in epistemology and philosophy of science, including learning, thought experiments, scientific modelling, and mathematics. The Routledge Handbook of Philosophy _of Imagination_ is essential reading for students and researchers in philosophy of mind and psychology, aesthetics, and ethics. It will also be a valuable resource for those in related disciplines such as psychology and art. (shrink)

In this book, I will discuss three main topics: the roots and aims of scientific knowledge, scientific knowledge in society, and science and values I understand scientific knowledge as being a planned and continuous production of the general and common knowledge of scientific communities. I begin my discussion with a brief analysis of the main differences between sciences, on the one hand, and everyday experience, philosophies, religions, and ideologies, on the other. I define the concept of science as a set (...) of family resemblances (in Wittgenstein's sense). The sciences can be distinguished along three lines, in terms of the difference between empirical and non-empirical sciences, differences in degrees of theoretical and methodological structure among individual sciences and scientific disciplines. Scientific knowledge is based upon some grounds, paradigmatic knowledge bases. Scientific knowledge is founded on a unified paradigmatic knowledge base. The latter includes fundamental theories, basic facts, generally accepted empirical regularities (laws), scientific research methods, scientific explanations, and established relations to other sciences. Scientific knowledge in general is defined well in classical terms by the definition that knowledge is a true and rationally-justified belief. In this connection, scientific truth is relative, depending on the given paradigmatic knowledge base and on the precision and accessibility of the measurements and observations involved. In the book's first part, I consider the production and testing of scientific hypotheses, scientific explanations, the structure of scientific theories, and the problem of scientific realism. I consider explanation an integral part of a wider nexus of heuristic and theoretical activities in science like giving reasons, understanding, clarification, and founding. Continuous production of scientific explanations is a necessary condition for the systematic reproduction of scientific knowledge. Accordingly, it does not suffice to describe them in terms of the logical structure of the arguments on which they are based, but rather, scientific explanations ought to be given complete with an epistemic and pragmatic context of the explanation. I consider especially deductive-nomological, inductive-statistical, dispositional, teleological-functional, and genetic explanations. I emphasize the ever-increasing importance of explanations by analogy and by model. Orthogonal to these distinctions lies the distinction between causal and epistemic explanation. Epistemic explanations relate scientific discoveries to epistemic progress, and causal explanations are these epistemic explanations that relate scientific discoveries to reality too. Causal explanations inform us that certain changes occur in the world as agents of other changes, or rather, that certain changes are real consequences of other changes. Thus, by having explanations of causes, we can intervene in the world to induce by design certain changes in some course of phenomena. I consider scientific theories as logically and epistemically ordered wholeness of scientific knowledge in a realm of research that enables us to construct useful models of reality. After the discussion of different 'statement' views on theories and different interpretations of the theory-experience relation, I introduce the 'model' theoretical (non-statement) views on theories, especially the structural theory of science. The model views on theories transcend the realism-antirealism dualism by introducing models between theory and reality. Therefore, the question of scientific realism is not merely whether and, if so, which element(s), of scientific theories represent reality, but rather which conditions suffice to infer from a theory that one of its models truly expresses reality. In other words, under which conditions do models of a theory represent models of reality? Generally, models are just as much images of reality in a theory as they are images of a theory in reality. The actual problem of scientific realism doesn't originate in the observable-unobservable distinction (even if this distinction is otherwise theoretically loaded), but rather in ontological quandaries, of the sort especially encountered in modern physical theories such as quantum mechanics, relativity theory, and cosmological theories, although they are known to arise in other sciences as well. Scientific theories may be real, even if they describe nothing about the world. However, such theories can be given this status only if their formal, mathematical postulates exhibit some formal homology between reality and mathematical forms. The very fact that it is possible to describe reality with the help of mathematics even in cases, where every (representational, conceptual) language and thought fall short, would indicate that even mathematics itself is, in some sense, a part of reality, although 'outside' of the empirical world. It follows we have to distinguish the non-empirical reality and the empirical world. Formal objectivity and non-arbitrariness of theories do not emerge from either our language or thoughts, or from the world of facts, causes, and effects, but rather from non-empirical reality, which frames the world as some kind of objective transcendental condition for the possibility of the existence of the world, thoughts, and language. In the second part of the book, I consider the incorporation of scientific work and scientific knowledge into modern society and its relations to values and ethics. Production of scientific knowledge goes on as a parallel distributed process of a special kind of social (shared) cognition. The central part of this cognition is the presentation of parts of the world by scientific (theoretical) models which are warranted by the shared and public practice of justification, and in reflective acting in the world as it would. I consider various formal concepts of collective propositional knowledge (distributed, mutual, common knowledge). Scientific knowledge is an achievement to be ascribed to a scientific community or, indirectly, to all those who apply or expand it, rather than to individuals per se. It presents the continuous and partially planned production of non-trivial common knowledge of the research communities in modern societies. However, besides common knowledge, shared dispositional knowledge (knowledge how of scientific teams) and shared knowledge by acquiring (knowledge of fundamental similarities and differences in scientific teams) are parts of the real body of scientific knowledge. Scientific common knowledge implies the projection into idealized universal knowledge of all real or potentially rational and well-informed individuals. The respect of scientific knowledge thus presents the norm of rationality. Science as the intensively shared and socially distributed kind of cognition enters into modern society as a special kind of work (general work), as a productive force (social invention drive), and as a form of capital (human, social, cultural, informational, etc.). However any of these categories can adequately represent the socio-economical role of science in modern societies, and we are still waiting for a more adequate general concept that would theoretically cover the 'scientification' of modern societies. My thesis is that the most important general effect of scientification is (and will be) the growth of social processes that structurally resemble parallel distributed shared social cognition of science, and not so much the rise of the use of scientific knowledge and scientifically based information in all kinds of production. Scientification of modern societies inevitably includes a much stronger need for moral and social responsibility of scientists for their work and its effects in their local society and the global world. This need comes in contradiction with the quite large accepted idea (and even the 'norm') on value-free research. I discuss to some degree different arguments for the absence of non-epistemic values and ethical reasoning from science and find them non-convincing. The sound idea of value-free research lies in the need for the absence of categorical value statements from scientific explanations and theoretical deductions. Values and ethical judgment have, and sometimes they even have to play an important role in another part of research and its application in science or outside it. Not the value-loudness of research is a danger for science but its blind, non-reflective use. It leads to the ideologization and politicization of science. A similar danger lies in the purely value-neutrality of research. It leads to the instrumentalization of science to the technical or bureaucratic means for 'any' aims. (shrink)

Ontic structural realism argues that structure is all there is. In (French, 2014) I argued for an ‘eliminativist’ version of this view, according to which the world should be conceived, metaphysically, as structure, and objects, at both the fundamental and ‘everyday’ levels, should be eliminated. This paper is a response to a number of profound concerns that have been raised, such as how we might distinguish between the kind of structure invoked by this view and mathematical structure in (...) general, how we should choose between eliminativist ontic structural realism and alternative metaphysical accounts such as dispositionalism, and how we should capture, in metaphysical terms, the relationship between structures and particles. In developing my response I shall touch on a number of broad issues, including the applicability of mathematics, the nature of representation and the relationship between metaphysics and science in general. (shrink)

In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in (...) class='Hi'>mathematical knowledge, in particular its dependence on mental/brain states and material objects. (shrink)

In a pair of very important papers, namely “Space, Time and Individuals” in the Journal of Philosophy for October 1955 and “The Indestructibility and Immutability of Substances” in Philosophical Studies for April 1956, Professor N. L. Wilson began something which badly needed beginning, namely the construction of a logically rigorous “substance-language” in which we talk about enduring and changing individuals as we do in common speech, as opposed to the “space-time” language favoured by very many mathematical logicians, perhaps most (...) notably by Quine. This enterprise of Wilson's is one with which I could hardly sympathize more heartily than I do; and one wishes for this logically rigorous “substance-language” not only when one is reading Quine but also when one is reading many other people. How fantastic it is, for instance, that Kotarbinski1 should call his metaphysics “Reism” when the very last kind of entity it has room for is things —instead of them it just has the world-lines or life-histories of things; “fourdimensional worms”, as Wilson says. Wilson, moreover, has at least one point of superiority to another rebel against space-time talk, P. F. Strawson; namely he does seriously attempt to meet formalism with formalism—to show that logical rigour is not a monopoly of the other side. At another point, however, Strawson seems to me to see further than Wilson; he is aware that substance-talk cannot be carried on without tenses, whereas Wilson tries to do without them. Wilson, in short, has indeed brought us out of Egypt; but as yet has us still wandering about the Sinai Peninsula; the Promised Land is a little further on than he has taken us. (shrink)

Gerald Holton has famously described Einstein’s career as a philosophical “pilgrimage”. Starting on “the historic ground” of Machian positivism and phenomenalism, following the completion of general relativity in late 1915, Einstein’s philosophy endured (a) a speculative turn: physical theorizing appears as ultimately a “pure mathematical construction” guided by faith in the simplicity of nature and (b) a realistic turn: science is “nothing more than a refinement ”of the everyday belief in the existence of mind-independent physical reality. Nevertheless, Einstein’s (...) class='Hi'>mathematical constructivism that supports his unified field theory program appears to be, at first sight, hardly compatible with the common sense realism with which he countered quantum theory. Thus, literature on Einstein’s philosophy of science has often struggled in finding the thread between ostensibly conflicting philosophical pronouncements. This paper supports the claim that Einstein’s dialog with Émile Meyerson from the mid 1920s till the early 1930s might be a neglected source to solve this riddle. According to Einstein, Meyerson shared (a) his belief in the independent existence of an external world and (b) his conviction that the latter can be grasped only by speculative means. Einstein could present his search for a unified field theory as a metaphysical-realistic program opposed to the positivistic-operationalist spirit of quantum mechanics. (shrink)

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. (...) In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

Shaffer takes a tour of some perennial questions in this lucid and simply written primer. How do I know I am not dreaming? How does reality differ from a dream? How can we be certain of our knowledge? Varying viewpoints are briefly summarized. The fallibilist view that even a priori mathematical truths and first person reports of feelings and perceptions are subject to error is examined, as is the anti-fallibilist reply that the theoretical possibility of error, without actual evidence, (...) is not sufficient to disturb the certainty of certain kinds of knowledge. Our knowledge of the external world is questioned by Berkeleyan idealism, which treats matter as nothing but a perception of our minds, and by phenomenalism, which treats material objects as nothing but collections of sense data. Opposed to these views are causal theories, which view physical objects as existing independently of observers, and sense data as the result of the causal interaction of material objects and particular bodies. It should be noted that the book's format requires such drastic foreshortening of Descartes' and Berkeley's views that its characterization of the former's system as "one of the great achievements in man's intellectual history" is in no way confirmed by the textual references. Other questions discussed in lively fashion are the self ; identity theory ; epiphenomenalism ; and interactionism. Is the mind best defined as the collection of mental events one has during a lifetime? If these events are all connected with the same body, the mind seemingly cannot survive the death of the body. Finally, theistic and atheistic views about the meaning of life are compared, with the Aristotelian view of teleological causation as the common denominator. The book asks many questions, provides few answers, but serves as an enticement to further exploration.--A. T. (shrink)

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained (...) logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives. (shrink)

Some recent discussions of A. N. Whitehead's treatment of the problem of value have stressed the point that his work in this field is open to serious objection. For example, Professor John Goheen claims that Whitehead's attempt to indicate distinguishing characteristics of experience of “the Good”, is too general to be adequate. He also suggests that this generality of approach makes it impossible for Whitehead to differentiate between different species of value. Further, according to Goheen, Whitehead involves himself in confusion (...) when he claims that satisfaction is achieved when experience is characterized by order ; and yet he also suggests that in some cases, the presence of disorder makes possible a higher type of satisfaction. Professor P. A. Schilpp agrees with Goheen in feeling that Whitehead's explanation of the good life in terms of “pattern” is too vague. In Schilpp's opinion, Whitehead does not answer the basic question: “What kind of pattern?” There is a further objection to Whitehead's “theory of the close connection between morality and beauty,” because at times it looks like an actual identification. Schilpp also objects to what appears to be Whitehead's identification of “good” with “interest”. Finally, it is claimed by Schilpp that Whitehead subordinates goodness to beauty. This interpretation is apparently supported by Prof. George Morgan who states: “Truth and moral values are instrumental except in so far as they enhance beauty.” Professor B. Morris objects to Whitehead's attempt to apply the mathematical method to aesthetic experience. He suggests that the method of symbolic logic is too abstract to deal adequately with concrete aesthetic data. Professor J. W. Blyth, as the result of his examination of Whitehead's theory of truth, reaches the conclusion that “the various elements involved in Whitehead's theory of truth cannot be brought together in one coherent and consistent system. (shrink)

For his contribution to the general series of Harper Essays in Philosophy, Hilary Putnam selects only one of several philosophical problems in the interrelated fields of logic and/or mathematics that have interested him, viz. the nominalism-realism issue: Are the "abstract entities" spoken of in these sciences, such as classes, number, functions from various kinds of things to real numbers, things that "really exist" or not? He is concerned to present a detailed argument for his own "qualified realism" rather than a (...) survey of current opinions. He argues persuasively that while a nominalist would like to eliminate all talk of logical classes such as S, M and P in favor of such circumlocutions as "The following schema turns out to be true no matter what words or phrases of the appropriate kind one may substitute for the letters S, M and P," etc., such descriptions avoid references to classes only by replacing them with equally nonphysical entities such as "all possible words, strings of letters, formalized languages," etc., and that the notion of truth or validity itself cannot be applied to physical objects or material inscriptions, but to what sentences mean or say, and the things they say are not physical. He rejects Quine's distinction that first order statements or arguments such as "All crows are black; all back things absorb light, therefore," etc., are logical truths, whereas "For all classes S, M and P, if all S is M and all M is P, then all S is P," are mathematical. This runs counter, he says, to the whole logical tradition. The natural intention of the logician in setting down first-order schemata is to assert implicitly their validity and that itself is a second-order assertion. All logic, therefore, including quantification theory, involves references to classes. This also highlights the difficulty of distinguishing between logic and mathematics in terms of first and second order logic, for if only the former is stipulated to be the proper domain of logic, then we must accept the awkward consequence that validity and implication turn out to be notions that belong to mathematics rather than logic. Since the nominalistic position presents the same problems for mathematics as for logic, Putnam sees no point in trying to draw any sharp distinction between the two, which explains his initial broad interpretation of logic as including all of pure mathematics and also the title of the book. To avoid some of the classical objections to a realistic interpretation of sets, he points up the distinction between a predicative and impredicative conception of set. Whereas the latter speaks of all sets of individuals as a well-defined totality, the former defines "set" only for a particular language N or a hierarchic series of languages, N, N', N", etc. in which it is still nonsensical to speak of "all sets" without qualification, though one can speak of "all" up to any given level in the series. While some form of set theory is necessary for all the sciences, both formal and physical, one can get along with predicative set theory. Putnam's basic argument for realism then comes down to this. Quantification over mathematical entities such as things, real numbers and functions is indispensable for logic, mathematics and the physical sciences and this commits one to accept the existence of such entities. In the final stages of the argument he examines and rejects the alternative options to realism, namely that realistic locutions about such entities represent logically deviant forms of speech, or that mathematical truths are "true by pure convention" or that such "entities" are useful fictions. In the closing paragraphs he indicates two other philosophical problems in logic that merit extensive discussion, namely the problem of equivalent constructions which is relevant to the question of whether mathematics needs "foundations" and the problem of whether we may not have to revise logical principles as we have geometrical ones on the grounds that in some sense logic may have empirical foundations.--A. B. W. (shrink)

This thesis examines necessary truth and defends a normative, or linguistic, account of it. Roughly, it holds that necessary truths state or follow from conceptual norms (i.e., norms that determine patterns of correct concept use). While the thesis touches upon logical and mathematical truth, its primary focus are those necessary truths typically expressed using natural language. The thesis has three parts. In Part I, I criticise metaphysical accounts of necessity and present and defend a normative account of it. At (...) no point do I give a history of normative accounts, but clearly their roots are to be found in the first half of the twentieth century – in the works of Wittgenstein and Carnap, for example. In Part II, I consider whether language can sustain the normative account. Some argue that the account requires language to be regimented in a way that it is not. I show that while it requires a distinction in kind between empirical and conceptual principles, it nevertheless makes room for indeterminacy regarding whether a given statement is an empirical claim or follows from conceptual norms. Finally, in Part III, I consider the relationship between the world and our conceptual scheme. I argue that denying our concepts answer to the world does not mean that they cannot be justified. The normative account does not say that we have no reasons for categorising things in a certain way, but rather that natural facts, in combination with our interests, are fit to provide them. The purpose of the thesis is to show that normative accounts of necessity can be much more robust than they are often given credit for and needn’t have the malign implications often associated with them. (shrink)

A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.

800x600 In our book _Global Insanity_ we argued that the existential predicament faced by humanity is a predictable consequence of Western Enlightenment thinking and the resulting world model, whose ascendance with the Industrial Revolution entrained development of the global consumer Economy that is destroying the biosphere. This situation extends from a dominant mindset based on the philosophy of reductionism. The problem was recognized and characterized by Robert M. Hutchins. In 1985, Hutchins ideas were discussed by Robert Rosen in Chapter 1 (...) of _Anticipatory Systems: Philosophical, Mathematical & Methodological Foundations_._ _Building on Hutchins' ideas, Rosen laid the foundation for an entire new, non-reductionist paradigm, which he called "complexity theory". This new paradigm is what we are further developing here. One has to recognize that a paradigm shift is needed to overcome the entrenched mindset and world model that reductionism has created. Here we explore the myriad interconnected ways-psychological, social, cultural, political, and technological-that the Western world model and consumer economy works as a complex system to thwart, neutralize, or co-opt for its own ends any effort to bring about the kind of radical change that is needed to avert global ecological catastrophe and societal collapse. This resistance to change stems from the need, inherent in the Western model, to continually grow the consumer economy. The media's continued portrayal of consumptive economic growth as a good thing, the widely held belief that the Economy is paramount, and current political and technological trends all manifest the system's active resistance to change. From the perspective of the mature economic system, any work that does not serve to grow the Economy is counterproductive, and viewed as unnecessary, a luxury, or subversive. The potential for real change is thus inversely related to the viability of the Economy, which will eventually decline as the system develops into senescence. To the extent that the fragile metastability of senescence affords opportunity for radical change, economic decline can be viewed as a hopeful sign. But taking maximum advantage of that opportunity will be extraordinarily difficult, as it will require widespread recognition of the problem, major voluntary sacrifice by the relatively large numbers of people who still benefit from the system, and concerted "grassroots" efforts. It can be expected that the system will resist those efforts until the end, becoming increasingly reliant on media-enabled distraction and divisive politics, as well as violent coercion, to maintain itself. Investment in education and science is widely touted as necessary for improving our situation, but this is misguided as long as the educational system and scientific enterprise continue to work in collusion with the larger system, as they currently do. Until the reductionist mindset and world model that drives the system is effectively challenged, there can be little hope for the kind of change needed to avert the catastrophic collapse of civilization. Normal 0 false false false EN-AU X-NONE X-NONE MicrosoftInternetExplorer4. (shrink)

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception (...) does not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field (with infinitesimals). Our approach was inspired by the work of (...) Whitehead (1919), though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

According to the realist rendering of mathematical structuralism, mathematical structures are ontologically prior to individual mathematical objects such as numbers and sets. Mathematical objects are merely positions in structures: their nature entirely consists in having the properties arising from the structure to which they belong. In this paper, I offer a bundle-theoretic account of this structuralist conception of mathematical objects: what we normally describe as an individual mathematical object is the mereological bundle of (...) its structural properties. An emerging picture is a version of mereological essentialism: the structural properties of a mathematical object, as a bundle, are the mereological parts of the bundle, which are possessed by it essentially. (shrink)

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called equation image without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in this (...) paper a Kripke-style semantics for a subsystem equation image of equation image for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system equation image with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic equation image. We introduce a reational semantics which has the advantage to allow an interpretation of the propositionnal logic equation image using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic equation image. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in (...) this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated (...) in this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic ${\rm PPC}_{3}$ , with two kinds of falsity and two negations. It introduces the notion of Σ-space for a sentence A (or / A /, the set of situations in which A is true) as the basis of logical model theory, and the notion of / ${\bf P}^{A}$ / (the Σ-space of the presuppositions of A), functioning as a 'private' subuniverse for / A /. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. ${\rm PPC}_{3}$ and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of ${\rm PPC}_{3}$ as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: $1_{a}$ for the conjoined presuppositions of a, ã for the complement of a within $1_{a}$ , and â for the complement of $1_{a}$ within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field. Our approach was inspired by the work of Whitehead, though (...) our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind (...) of view: its proponents rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” (...) or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics. This I will explain below. The second statement expresses what looks like a rather wild thesis. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:John Davis, Teacher: A Recollection I first met John Davis in the late fifties when I was doing a two-year M.A. in philosophy at Western. John taught a graduate course in symboliclogic. It was both a philosophy course and across-listed course in the foundation ofmathematics. There was, as a result, a mixture of philosophy and mathematics students in the course. My most vivid memory ofJohn was ofan amazingly energetic (...) and enthusiastic teacher who was anything but indifferent to his subject matter and to his students. I, if I may introduce a personal note, suffered from an extreme form of math anxiety—I couldn't even understand a truth-table. John was always extremely patient with me and it is thanks to him that I managed to scrape through. Another vivid memory was of a man with a kind of Moorean passion for getting it right. When he didn't understand something, he let us know it. He was constitutionally incapable of intellectual pretence. I took away from my time with him the idea that philosophy was extremely difficult; but ifone worked very hard, it was possible to get the right answer. And no relativist he, John was profoundly convinced that there was a right answer to get. Robert A. lmlay... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.2 (2003) 275-276 [Access article in PDF] Zbigniew Janowski. Cartesian Theodicy: Descartes' Quest for Certitude. Dordrecht: Kluwer, 2002. Pp. 181. Cloth, $30.00. Janowski begins this original and erudite work by saying that although "the Meditations have never [before] been interpreted as a theodicy... insofar as theodicy is concerned with examining the relationship between the existence of evil on the one hand and God's (...) omnipotence and benevolence on the other, Descartes's question 'How would the goodness of God not preclude the possibility that nature is deceptive?' is in perfect conformity with, and continues the long tradition of, Christian apologetics" (13). This is not to deny Descartes's primary role in replacing God's causal role in nature with mechanistic science, and his separating religious truth from that of reason. But Descartes also is profoundly concerned with problems of error, evil, God's role in our sins, and Absolute Truth. Janowski is concerned "to show how Descartes' philosophy is derived from the early-seventeenth-century debates over divine and human freedom" by concentrating on the "incongruity between the existence of an omnipotent and benevolent Creator and man's imperfect nature [and] the nature and scope of human freedom" (19). He presents "the problem of error... as the problem of evil; the problem of human will... as the problem of human freedom, and the quest for Certitude... as theodicy, that is, as the vindication of God's goodness and omnipotence" (19-20). His ground for this is his interpretation of Cartesianism as "a kind of epistemological Augustinianism" (20).The view of basic Christian theology is that God is omnipotent and benevolent, and that all his creations are good, so evil is merely the absence of good. According to Descartes, free will makes both error and evil possible. But Augustine says that after the Fall of Man, free will is corrupt and can incline only to evil. So the only good man can do is by way of God's grace, which man cannot resist, for man is not really free to choose between good and evil. But here, Descartes is in disagreement with Augustine. Again, theodicy concerns the problem of the apparent discrepancy between the power of the creator and the imperfections of creation. And the problem is that Descartes's God is an all-powerful being that is absolutely unlimited and creates everything including logical, moral, and legal principles. Descartes's God is not constrained by the principles of non-contradiction (as opposed to St. Thomas's God, who is so constrained). Descartes's argument is that only God is immutable, nothing created is immutable, so no principle created by God, not the Ten Commandments, not the principle of non-contradiction, is immutable.This doctrine of indifferently created eternal truths allies Descartes with the Molinists who argue that God is indifferent in the sense of all powerful, and that humans have a freedom of indifference in that even given grace, they can decide to do either good or evil. This is in opposition to Augustinians and Jansenists who believe that human will corrupted by the Fall of Man can choose only evil until God gives one grace, after which one can choose only good—which, in either case, rules out any notion of freedom of choice between good and evil by an indifferent free will. Descartes believes that we can choose evil even when God gives us grace to choose good. Descartes does follow Augustine in asserting that for God, knowing and willing are the same, in opposition to St. Thomas. But for humans, knowing and willing are separate, and so humans can both err and sin.Descartes's theodicy is that everything depends on God—physics, mathematics, morals, and law. Without God there would be only nihilism. A great difficulty with this doctrine of eternal truths, however, is that it "makes the nature of God completely incomprehensible to man" (102). This is the doctrine of the Hidden God. But in a backhanded way, it saves Descartes's position. He argues that... (shrink)

It is our contention that an ontological commitment to propositions faces a number of problems; so many, in fact, that an attitude of realism towards propositions—understood the usual “platonistic” way, as a kind of mind- and language-independent abstract entity—is ultimately untenable. The particular worries about propositions that marshal parallel problems that Paul Benacerraf has raised for mathematical platonists. At the same time, the utility of “proposition-talk”—indeed, the apparent linguistic commitment evident in our use of 'that'-clauses (in offering explanations (...) and making predictions)—is also in need of explanation. We account for this with a fictionalist analysis of our use of 'that'-clauses. Our account avoids certain problems that arise for the usual error-theoretic versions of fictionalism because we apply the notion of semantic pretense to develop an alternative, pretense-involving, non-error-theoretic, fictionalist account of proposition-talk. (shrink)

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.

The social welfare functional approach to social choice theory fails to distinguish a genuine change in individual well-beings from a merely representational change due to the use of different measurement scales. A generalization of the concept of a social welfare functional is introduced that explicitly takes account of the scales that are used to measure well-beings so as to distinguish between these two kinds of changes. This generalization of the standard theoretical framework results in a more satisfactory formulation of welfarism, (...) the doctrine that social alternatives are evaluated and socially ranked solely in terms of the well-beings of the relevant individuals. This scale-dependent form of welfarism is axiomatized using this framework. The implications of this approach for characterizing classes of social welfare orderings are also considered. (shrink)

One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted (...) to elementary arithmetic, it requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence. (shrink)

This is a translation of a work, which appeared originally in French in 1923, that exposes in considerable detail the doctrine of Giordano Bruno on various kinds of minima. Bruno is justly famous for his teachings on infinity, but is little known for adumbrating atomic concepts through his finalist approach to the ultimate constituents of matter and mathematical continua. The author proposes to remedy this defect by exposing, in a systematic way, the contents of Bruno’s somewhat confusing Latin poem (...) De triplici minimo et mensura, ad trium speculativarum scientiarum et multarum activarum artium principia, first published in 1591 at Frankfurt. Like others of Bruno’s compositions, this is "a tiresome melange of irregular hexameters," in imitation of Lucretius, interspersed with Latin scholia "occasionally more poetic than the verses themselves". In three chapters the author successively details the various forerunners of Bruno’s doctrine ; makes sense of the three minima ; and offers a critique. (shrink)

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...) of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:31 Hunie on Intuitive and Demonstrative Inference This paper is divided into four sections. The first section deals with Hume's attempt to resolve a dilemma concerning the objects of intuitive and demonstrative inference. In the second section I try to show that his resolution of the dilemma is hard to reconcile with his phenomenalist doctrine of the origin of ideas. In the third section I examine tne meaning of (...) "intuition" in Hume. Finally, in the fourth section I compare his view of mathematical inference with his view of causal inference, a kind of inference to which he tends to assimilate the former. What are the objects of intuitive and demonstrative inference in Hume supposed to be? In the Treatise they are co-extensive with four kinds of philosophical relations resemblance, contrariety, degrees in quality and proportions in quantity or numbers. The distinguishing characteristic of the group, according to Hume, is that they all depend entirely on the ideas which we compare together. Tne example he gives is that of a triangle 2 whose three angles are equal to two right ones. And this presumably is at the same time an example of a relation of proportion in quantity or numbers. Needless to say, what cries out for explanation here, apart from the notion of comparison with which we shall deal later, is the phrase "depend entirely upon the ideas." Hume himself does not provide the explanation. Consequently, a certain amount of speculation becomes inevitable. One fairly plausible although, as we shall see, not entirely satisfactory interpretation of the phrase in question would associate it with a thesis concerning the reducibility of relations between exemplars of properties to relations between the properties themselves. Thus it may 32 be argued that the equality holding between the three angles of a triangle and two right angles is reducible to an equality holding between certain of their properties. The properties would be the 180° that each set of angles exemplifies. Indirect evidence for this interpretation is provided by Hume's account of the relations of contiguity and distance, species presumably of relations of time and place, which do not, according to him, depend upon the ideas wnich are compared together. Indeed, these relations may be changed by "an alteration of their place, without any change on the objects themselves or on their 3 ideas." "The place" in its turn "depends on a hundred 4 different accidents which cannot be foreseen by the mind." Leaving aside Hume's failure or refusal to draw a clear distinction here between ideas and objects which probably has to do with his belated realization that only the latter have places and are contiguous to or distant from one another, one notes his insistence that an object's place is not an essential property of the object. For his use of the word "accident" in the above-quoted passage has very little to do with its ordinary use and everything to do with its logical one where it is contrasted with "essential." Furthermore, the sentence immediately preceding the one containing that passage gives some indication of what Hume takes its logical use to be : A determinate property is an accidental one if loss of that property does not affect the object's identity or, as he would put it, the object itself. Nor is the accidental nature of the property anything but reinforced by its dependence logical or otherwise upon other accidental properties which like all such properties require an appeal to observation and induction in order to determine whether they are exemplified by that object. The loss of an essential property, on the other hand, would affect the object's identity because such properties are contained in the very definition of the object. And it suffices to understand the definition - Hume's foreseeing by the 33 7 mind comes in here - in order to determine that the property is exemplified by the object. How does this account provide indirect evidence for our interpretation of Hume? The answer is relatively straightforward. According to that interpretation, relations which depend entirely on the ideas which we compare together are tnose where relations between exemplars of properties are reducible to relations between... (shrink)

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant (...) Bermúdez’s version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (shrink)

Despite some surface similarities, Charles Peirce’s philosophy of mathematics, pragmaticism, is incompatible with both mathematical structuralism and fictionalism. Pragmaticism has to do with experimentation and observation concerning the forms of relations in diagrammatic and iconic representations ofmathematical entities. It does not presuppose mathematical foundations although it has these representations as its objects of study. But these objects do have a reality which structuralism and fictionalism deny.

For Miss Sewell our apprehension of the world is basically through myth. Art, language, and even mathematics, rightly understood, are kinds of myth. This book centers upon those poets and biologists who share common goals by virtue of their use of the primary form of myth, i.e., "world-language." The major part of this book deals in these terms with such thinkers as Bacon, Linnaeus, and Rilke.--J. A. B.

The five chapters in this volume were originally delivered in lecture form at the University of Genoa and have previously appeared in French, German, and English translations. An appendix, "What the Philosopher May Learn from the Study of Law," has also appeared before in English. The book is basically a digest, with some modifications, of Perelman's earlier work Justice et Raison. The chief modification involves a supposed shift away from positivism toward a greater emphasis on the cognitive status of primary (...) values. Through the mediation of his consideration of the nature of practical reason in the Traité de l'Argumentation, Perelman now wishes to argue that values are subject to dialectical justification. The analysis of justice, thus, need not be restricted to merely formal specifications such as the rule of justice or to such inherently nonuniversalizable and nonanalyzable principles as that of equity. Dialectical justification falls short of demonstration, but only someone still under the spell of Cartesianism would wish to maintain that rationality is adequately summarized in the model provided by mathematics and science. Jurisprudence, therefore deals with judgments that have cognitive and not arbitrary or merely historical value. So much for the scheme of Perelman's argument. What about its execution? It must be said that its substance is slight and the base of discussion extraordinarily narrow. Thus, Perelman says he wishes to avoid the Scylla of nominalism and the Charybdis of realism; but his own analysis winds up squarely in a conventionalism that is only minutely distinguishable from the nominalism he presumably attacks: e.g., "the idea of [practical] reason must not be linked to that of truth", "reason is but an attempt to convince the members of [the universal] audience--whom common sense would define as well-informed and reasonable men". The only defense offered by Perelman for the first statement is a flaunting of the bogey of dogmatism. The vicious circle in the second statement is barred from rising to the level of the fruitful Aristotelian paradox by a peremptory dismissal of natural law: "The theorists of natural law... have not convinced the majority of those philosophers of law who are endowed with critical minds." This kind of "you-don't-belong-to-the-club" argument makes the book practically worthless from a truly dialectical point of view. The only critical sparks generated are in Perelman's discussion of Rawls' theory of "justice as fair play." His points here are well taken. But, as this is the only saving grace in the book--the rest being merely a statement of Perelman's own doxa about justice--it would have been better to leave the whole interred in the journals where it had already been published.--E. A. R. (shrink)

Salutary reading for all philosophers, and not only for inductive logicians, philosophers of science and law, this important book presents an elaborate theory of inductive reasoning whose substantive features are as strikingly original as the approach is rare. First, the theory is based on concrete, real, actual, and significant instances of inductive reasoning, e.g., Karl von Frisch’s work on bees; that is, though its aim is genuinely theoretical in the sense that it engages in the proper amounts of idealization, abstraction, (...) systematization, precision, and rigor, it never loses sight of the fact that "theories in the philosophy of science, like theories in science itself, have to face up to the challenge of appropriately accredited experience within their appointed domains". Second, a central subject matter being analyzed is legal reasoning, that is, the types of proofs and arguments used by juries, judges, and lawyers in the Anglo-American system of jurisprudence, where criminal charges have to be proved beyond reasonable doubt and civil cases have to be argued on the balance of probability. Thus, in effect, Cohen’s synthesis of juridical and of scientific reasoning is an admirable example of the bridging of two "cultures." Third, and most importantly, the concept of "inductive probability," in terms of which Cohen makes sense out of his subject matter, is incommensurable with that of "mathematical probability," namely the classical calculus of chance according to which probability is measurable, additive, transitive, and obeys a multiplicational principle for conjunction and a complementational one for negation. This is not to say that "inductive probability" is a hazy, mysterious, or unstructured concept, and Cohen goes to great lengths to show that it is ordinal, modal, and formally analyzable; what it does mean is that essential types of reasoning in science and in law involve probable inferences whose probabilities are nonquantitative. Hence, though Cohen does not stress the qualitative nature of inductive probability, his book will be welcome by those who conceive of philosophy as scientia qualitatum. Specific conclusions reached by Cohen are that, in general, probability may be conceived as degree of provability, that differences in proof-rules generate different special cases of probability, that inductive probability may be viewed as the special case where the proof-rules are incomplete, that inductive probability is to inductive support as deducibility is to logical truth, and that some recent epistemological scepticism may be refuted by Cohen’s inductive logic. Finally, he sees himself in the British inductivist tradition, interpreting his predecessors as groping toward a logic of inductive support and probability rather than a methodology of discovery: from Bacon he develops the notion of eliminative induction but rejects that of induction by enumeration; Mill’s canons are analyzed as special cases of Cohen’s own "method of relevant variables," except for the method of residues, which is rejected as invalid ; he takes his critique of mathematical probability as a vindication of Hume’s thesis that "probability or reasoning from conjecture may be divided into two kinds, viz., that which is founded on chance and that which arises from causes" ; Whewell’s consilience of induction is justified ; and from Keynes, he develops the notion of the "weight" of evidence but rejects his mathematicist approach.—M. A. F. (shrink)

The five chapters in this volume were originally delivered in lecture form at the University of Genoa and have previously appeared in French, German, and English translations. An appendix, "What the Philosopher May Learn from the Study of Law," has also appeared before in English. The book is basically a digest, with some modifications, of Perelman's earlier work Justice et Raison. The chief modification involves a supposed shift away from positivism toward a greater emphasis on the cognitive status of primary (...) values. Through the mediation of his consideration of the nature of practical reason in the Traité de l'Argumentation, Perelman now wishes to argue that values are subject to dialectical justification. The analysis of justice, thus, need not be restricted to merely formal specifications such as the rule of justice or to such inherently nonuniversalizable and nonanalyzable principles as that of equity. Dialectical justification falls short of demonstration, but only someone still under the spell of Cartesianism would wish to maintain that rationality is adequately summarized in the model provided by mathematics and science. Jurisprudence, therefore deals with judgments that have cognitive and not arbitrary or merely historical value. So much for the scheme of Perelman's argument. What about its execution? It must be said that its substance is slight and the base of discussion extraordinarily narrow. Thus, Perelman says he wishes to avoid the Scylla of nominalism and the Charybdis of realism; but his own analysis winds up squarely in a conventionalism that is only minutely distinguishable from the nominalism he presumably attacks: e.g., "the idea of [practical] reason must not be linked to that of truth", "reason is but an attempt to convince the members of [the universal] audience--whom common sense would define as well-informed and reasonable men". The only defense offered by Perelman for the first statement is a flaunting of the bogey of dogmatism. The vicious circle in the second statement is barred from rising to the level of the fruitful Aristotelian paradox by a peremptory dismissal of natural law: "The theorists of natural law... have not convinced the majority of those philosophers of law who are endowed with critical minds." This kind of "you-don't-belong-to-the-club" argument makes the book practically worthless from a truly dialectical point of view. The only critical sparks generated are in Perelman's discussion of Rawls' theory of "justice as fair play." His points here are well taken. But, as this is the only saving grace in the book--the rest being merely a statement of Perelman's own doxa about justice--it would have been better to leave the whole interred in the journals where it had already been published.--E. A. R. (shrink)

The paper proposes to amend structuralism in mathematics by saying what places in a structure and thus mathematical objects are. They are the objects of the canonical system realizing a categorical structure, where that canonical system is a minimal system in a specific essentialistic sense. It would thus be a basic ontological axiom that such a canonical system always exists. This way of conceiving mathematical objects is underscored by a defense of an essentialistic version of Leibniz’ principle (...) according to which each object is uniquely characterized by its proper and possibly relational essence (where “proper” means “not referring to identity"). (shrink)

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and (...) its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

The urgency of the study of "stem-education" as a factor in the development of "smart-society" is that this kind of society is a continuation of information and "knowledge society", which is developing on the basis of smart technologies. The concept of smart society is at the heart of modern state -owned development programs of South Korea and Japan. In South Korea, the National Social Agency has developed a "Smart Society Strategy" that introduces the technological foundations of smart societies. The (...) central issue of Smart-society is business, which makes managing more intelligent and activities aimed at using knowledge and innovation. The objectives of the study are to identify the "stem-education" endpoint through the use of electronic and collective technologies, which become more massive and effective, and through the use of natural, engineering and maternal education. It is no coincidence that this concept is documented in the Europe 2020 Strategy: Smart Growth Strategy, which includes the development of an economy based on knowledge and innovation, fosters sustainable growth, more efficient use of resources, and inclusive growth and strengthening of high employment. Countries such as the United States, China, Australia, the United Kingdom, Israel, and Singapore carry out state-of-the-art stem-education programs. The methodological basis of the study is systematic, structural, axiological, synergetic methods and approaches that have allowed disclosing stem -education as the leading trend in the world of educational space. Conclusion - It is proved that stem-education is an innovation that combines the traditions of natural and mathematical education, is based on the principles of fundamentalism and knowledge -based, combining technological, organizational, material and technical resources and human capital. As a result of the development of stem-education at the expense of ICT, business processes, governance, and management reform are undergoing change, leading to economic, social and managerial processes at a higher level in some of those societies. Smart society is a new quality of society in which the combination of the use of new technologies will allow people to improve the quality of their lives. This article presents philosophical and educational reflection of smart -society as a new model of the information society and presents its impact on human capital. It reveals timeliness of this topic, which is innovative and hardly developed. It analyses international experience in establishment and growth of smart -society and dimensions of axiological field of smart-society, which is based on axiological matrix of information and knowledge, which are considered and being civilized dimensions of modern society. The main idea is to prove the evolution of the information society to smart -society and the possibility of establishment of smart-society in Ukraine. The analysis of smart-society formation was made and its characteristics were defined, which claims priority role in the world information space formation and contribute to the competitiveness of Ukraine in the international information space. (shrink)

An affirmative answer is given to the question quoted in the title.

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it (...) is argued that the modality of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

Modality - the question of what is possible and what is necessary - is a fundamental area of philosophy and philosophical research. The Routledge Handbook of Modality is an outstanding reference source to the key topics, problems and debates in this exciting subject and is the first collection of its kind. Comprising thirty-five chapters by a team of international contributors the Handbook is divided into seven clear parts: worlds and modality essentialism, ontological dependence, and modality modal anti-realism epistemology of (...) modality modality in science modality in logic and mathematics modality in the history of philosophy. Within these sections the central issues, debates and problems are examined, including possible worlds, essentialism, counterfactuals, ontological dependence, modal fictionalism, deflationism, the integration challenge, conceivability, a priori knowledge, laws of nature, natural kinds, and logical necessity. The Routledge Handbook of Modality is essential reading for students and researchers in epistemology, metaphysics and philosophy of language. It will also be very useful for those in related fields in philosophy such as philosophy of mathematics, logic and philosophy of science. (shrink)

In this journal (Studia Logica), D. Rizza [2010: 176] expounded a solution of what he called “the indiscernibility problem for ante rem structuralism”, which is the problem to make sense of the presence, in structures, of objects that are indiscernible yet distinct, by only appealing to what that structure provides. We argue that Rizza’s solution is circular and expound a different solution that not only solves the problem for completely extensive structures, treated by Rizza, but for nearly (but not) (...) all mathematical structures. (shrink)

A number of authors have recently argued that the mathematical insights of "chaos theory" offer a promising formal model or significant analogy for understanding at least some historical events. We examine a representative claim of each kind regarding the application of chaos theory to problems of historical explanation. We identify two lines of argument. One we term the Causal Thesis, which states that chaos theory may be used to plausibly model, and so explain, historical events. The other we (...) term the Convergence Thesis, which holds that, once the analogy between history and chaos theory is properly appreciated, any temptation to divide history from the rest of science should be greatly lessened. We argue that the proffered analogy between chaos theory and history falls apart upon closer analysis. The promised benefits of chaos theory vis-à-vis history are either fantastic or, at best, an extremely loose heuristic which, while retaining nothing of the considerable intrinsic interest of nonlinear dynamics, easily seduces the unwary into taking at face value terms and concepts that have a specifically precise meaning only within the confines of mathematical theory. (shrink)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the (...) main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that equation image. Note that equation image is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate equation image by μ. It is known that whenever A so presents α then A ≤wttα, where ≤wtt (...) denotes weak truth table reducibility, and that the wtt-degrees of presentations form an ideal ℐ in the computably enumerable wtt-degrees. We prove that any such ideal is equation image, and conversely that if ℐ is any nonempty equation image ideal in the computably enumerable wtt-degrees then there is a computable enumerable real α such that ℐ = ℐ. We also prove a kind of Rice Theorem for these ideals, namely that if the index set of such a equation image ideal is not empty or equal to ω then it is equation image-complete. (shrink)