This chapter contains sections titled: Introduction Platonism in Mathematics Nominalism in Mathematics Conclusion References.

Article is devoted to the social legitimation of knowledge. We study the contexts of implantation of knowledge products into the body of culture. The author proceeds from the need to study the process of objectification symbolic of object by applying the category of ‘facies‘, the introduction and justification of which on content and formal level were realized by the author in previous works. Such issues as the following are discussed in the article: the main stages of objectification, (...) cognitions, different worlds in science in the trend of unfolding cognitions, the basic principles of object-symbolic interpretation of the nature of knowledge. The author explores these questions by analyzing the basic epistemological trends that can be seen in the works of the greatest philosophers of both modern and recent times, globalizing the principle of a system approach on the basis of its synthesis with the elements of synergetics and semiotics. The author devotes a great place to criticism of Kant’s transcendental apriorism, revealing, however, its positive potential and determining its promising vector beating literally in the essence of contemporary cognitive science. Author takes on orbit of its original methodology conclusions of not only natural sciences and mathematics, but also poetic insights that allows creating of pulsating and at the same time quite certain interdisciplinary context. The main conclusions regarding the content of prerequisites of cognitive synthesis included in the final paragraph of article and are accompanied by convincing examples of modern physics. (shrink)

Kitcher's philosophical approach has moved from the reflection on the nature of mathematical knowledge to an explicit social concern about science, because he considers seriously the relevance of democratic values to scientific activity. Focal issues in this trajectory - from the internal perspective to the external - have been naturalism and scientific progress, which includes studies of the uses of scientific findings in the social milieu. Within this intellectual context, the chapter pays particular attention to his (...) epistemological and methodological evolution. The analysis of Philip Kitcher's contents on progress begins with mathematics, a conception that follows a naturalist perspective. Thereafter, the growth of science comes to the front line, an advancement that he views according to realism and cognitive naturalism. Later, the social concern about science receives a visible consideration, when his vision of scientific undertaking is characterized following modest realism and social naturalism. After these four steps (philosophical context, progress in mathematics, the growth in science, and the social concern about science), there is an analysis of his philosophical-methodological framework in retrospective. This is continued by the presentation of the origins of this book and the bibliography related to this thinker. (shrink)

The concept of complexity has become very important in theoretical biology. It is a many faceted concept and too new and ill defined to have a universally accepted meaning. This review examines the development of this concept from the point of view of its usefulness as a criteria for the study of living systems to see what it has to offer as a new approach. In particular, one definition of complexity has been put forth which has the necessary precision and (...) rigor to be considered as a useful categorization of systems, especially as it pertains to those we call living. This definition, due to Robert Rosen, has been developed in a number of works and involves some deep new concepts about the way we view systems. In particular, it focuses on the way we view the world and actually practice science through the use of the modelling relation. This mathematical object models the process by which we assign meaning to the world we perceive. By using the modelling relation, it is possible to identify the subjective nature of our practices and deal with this issue explicitly. By so doing, it becomes clear that our notion of complexity and especially its most popular manifestations, is in large part a product of the historical processes which lead to the present state of scientific epistemology. In particular, it is a reaction to the reductionist/mechanistic view of nature which can be termed the Newtonian Paradigm. This approach to epistemology has dominated for so long that its use as a model has become implicit in most of what we do in and out of science. The alternative to this approach is examined and related to the special definition of complexity given by Rosen. Some historical examples are used to emphasize the dependence of our view of what is complex in a popular sense on the ever changing state of our knowledge. The role of some popular concepts such as chaotic dynamics are examined in this context. The fields of artificial life and related areas are also viewed from the perspective of this rigorous view of complexity and found lacking. The notion that in some way life exists at the edge of chaos is examined from the perspective of the second law of thermodynamics given by Schneider and Kay. Finally, the causal elements in complex systems are explored in relation to complexity. Rosen has shown that a clear difference in causal relations exists between complex and simple systems and that this difference leads to a uniquely useful definition of what we mean by living. Rosen makes it very clear that the class of systems which are complex is a much larger class than those which we call living. For that reason, the focus of this review will be on complexity as a stepping stone towards the deeper question of what makes a system alive. (shrink)

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of (...) mathematics as a community of practice and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections (...) on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour. (shrink)

Werkmeister's new book, The Basis and Structure of Knowledge is the second major attempt in recent years to defend the idealistic theory of knowledge. The first was Blanshard's Nature of Thought; and it is worth noticing that both authors, in undertaking the defense of a position long in the shadows, are well aware of contemporary developments in logic and technical philosophy. Werkmeister freely acknowledges his debt to Blanshard; yet his work differs in scope from the latter's in (...) at least two ways. Firstly, the more recent book is concerned largely with the epistemological analysis of meaning and truth and hardly at all with what an older generation called "philosophical psychology," while philosophical psychology fills all of the first and part of the second volume of Blanshard's work. In my opinion, this is all to the good. (That our generation inverts the error and too often indulges in psychological philosophy is another story!) Secondly, Werkmeister presents us with detailed analyses of scientific method, of the structure of scientific theories, and of mathematics, all three areas to which Blanshard pays virtually no attention. The First Part of Werkmeister's book treats of Language and Meaning; the Second of Truth and the World About Us, the Third of Formal Knowledge, and the Fourth Part deals with Empirical Knowledge. But there is a fundamental thesis that runs through all of this. The analysis of each of these four domains reveals, according to Werkmeister, that only a coherence theory of meaning and truth can adequately reflect the criteria actually used by science and commonsense. It is not my intention to rehearse here the familiar dialectic of the arguments for and against this doctrine; the less so, since Werkmeister himself gives a useful review of both sides of the controversy. Rather, I want to try to fix as precisely as possible, within the context of Werkmeister's discussion, the nature of the difference between coherence theories on the one hand and correspondence theories on the other. This difference has not always been too clearly stated; it even seems that recent developments in linguistic analysis have been the occasion for some new confusions on this very point. For coherence theorists who have recognized that philosophical semantics is a "formal" discipline now insist that "to say what a symbol means is not to relate it to an object, but to give it an interpretation in terms of other symbols," as Ayer2 has recently put it. This seems to contradict the "comparison with fact" feature of the traditional realistic formulations of the correspondence view. A superficial bystander could even believe that the correspondence theorists have given up their original position, since both sides now agree that truth is a matter of a relation between symbols. Such agreement, as I hope to show, is more apparent than real, for what is now at stake is the nature of that "intrasymbolic relation." And on this nature the two parties disagree. There is little doubt that Werkmeister's book appears at a moment more propitious for the advocacy of the coherence theory of truth than did Blanshard's. In the near decade that separates the two works the intellectual climate has become, for better or for worse, considerably more favorable to a reconsideration of, and possibly even a reconciliation with, some of the major tenets of rationalism. This is reflected in the current discussion regarding the contrary-to-fact conditional, the dissatisfaction with material implication, and the correlative concern for a more adequate analysis of the concept of a natural law. It is evident too in the revived enthusiasm for intensional logic and in the related interest in the modalities. As uncompromising an empiricist as C. L. Lewis, with whom, by the way, Werkmeister shares a common Kantian background, has introduced into his latest work a "congruence" theory of truth which, despite the substitution of a so-called "probability" inference for necessary inference, looks very much like the coherance criterion all over again. So Werkmeister's book should be welcome for its clear presentation of the relevant issues; and even those who disagree with him can admire the candor and forthrightness with which he champions what remains, despite vacillations, an essentially unpopular position. (shrink)

There is an as yet unacknowledged and incomparable contribution to the philosophical debates about know-how to be found in the writings of Ludwig Wittgenstein. It is sourced in his investigations into knowledge and certainty in On Certainty, though it is not limited to these late passages. Understanding the ramifications of this putative contribution (even if one does not agree with it) highlights the extent to which (i) there is now a new range of issues pertaining to know-how (...) which no future philosophical consideration of the topic can ignore, except on pain of failing to engage comprehensively with the subject; (ii) the topic of know-how has been inappropriately marginalised by naturalized epistemology, and may well be as central to epistemology as the propositional knowledge which currently dominates epistemology’s attention; and (iii) any engagement with these potential Wittgensteinian contributions will need to be conducted in tandem with a reflection on the meta-philosophy of epistemology, since their potential impact extends to epistemology’s main methodology, i.e., naturalized reflective equilibrium. These three conclusions, together with a diagnosis of where and why all the current intellectualist accounts of know-how are either internally inconsistent, or irreconcilably flawed on their own terms, provide the motivation and the opportunity for a New Epistemology of Know-How. These conclusions established, I offer one possible Wittgensteinian-orientated version of the New Epistemology of Know-How, providing the first example of a non-naturalized philosophical approach to the topic since Gilbert Ryle. (shrink)

Although knowledge is a central topic for MKM there is little explicit discussion on what ‘knowledge’ might actually be. There are specific intuitions about form and content of knowledge, about its structure, and epistemological nature that shape the MKM systems, but a conceptual model is missing. In this paper we try to rationalize this discussion to give MKM a firmer footing, to start a discussion among MKM researchers and help relate the MKM intuitions and discourses (...) to other communities. Starting from the observation that many concrete realizations of mathematical knowledge objects are considered equivalent, we propose a conceptual model of the space of (mathematical) knowledge objects graded by levels of abstraction and presentational explicitness and draw conclusions for MKM markup formats. (shrink)

IntroductionScientific pluralism is generally understood in the backdrop of scientific monism. So is mathematical pluralism. Though there are many culture-dependent mathematical practices, mathematical concepts and theories are generally taken to be culture invariant. We would like to explore in this paper whether mathematical pluralism is admissible or not.Materials and methodsMathematical pluralism may be approached at least from five different perspectives. 1. Foundational: The view would claim that different issues within mathematics need support of different foundations, (...) apparently incompatible with one another. 2. Ontological: The world itself is dappled—the mathematical counterpart of which can be traced to the admission of non-Euclidean spaces and also in simultaneous acceptance of set-theoretic and category theoretic entities in the ontology of mathematics. 3. The third route is epistemological. It follows from the view that the nature of reality is so complex that different aspects of it require alternative modes of representation and explanation sometimes severally, sometimes simultaneously, e.g., classical, constructivist, computer-aided and different finitist mathematics can be used depending on the knowledge situation. 4. The fourth route is semantic. Fuzzy mathematics, we know, gives up bivalence in theory of truth, and ascribes truth to mathematical propositions in degrees. 5. The last one is Programmatic: for some, mathematical pluralism is a stance, a programme. It provides a framework which strikes at the root of cultural hegemony and nourishes a climate of intellectual humility and tolerance. It is not my intention to hold the brief of any of these versions. I just want to present to the readers a more or less complete picture of the scenario, though not an exhaustive one. ConclusionUnlike monism, mathematical pluralism does not rule out any possibility—even the possibility of having a unitary over-arching framework of interpretation remains as an open option. This paper highlights the fact that motivations for upholding pluralism in mathematics have been many and one can be a pluralist just by subscribing to any of the views listed above. (shrink)

Increasing attention has been given to “indigenous” knowledge in Third World rural societies as a potential basis for sustainable agricultural development. It has been found that many people have functional knowledge systems pertaining to their resources and environment, which are based on experience and experimentation, and which are sometimes based on unique epistemologies. Efforts have been made to include such knowledge in participatory research and projects. This paper discusses socio-political, institutional, and ethical issues that need (...) to be considered in order to understand the actual limitations and contributions of such knowledge systems. It reviews the nature of local knowledge and suggests the need to recognize its unique values yet avoid romanticized views of its potential. Local knowledge and alternative bottom-up projects continue to be marginalized because of the dominance of conventional top-down R&approaches, pressures of agrochemical firms, scientific professionalism, and for other political-economic reasons. It is argued that the exploitation of local knowledge by formal institutions should be avoided; instead, people need to establish legitimacy of their knowledge for themselves, as a form of empowerment. (shrink)

Scientific perspectivism, or perspectival realism, is a view according to which scientific knowledge is neither utterly objective nor independent of the world “as it is”, but always tied to some particular ways of conceptualization and interaction with Nature. In the present paper, I employ Robert Rosen’s concept of the modeling relation for arguing that there are two basic reasons why our knowledge of natural systems is perspectival in this sense. The first of these pertains to the dualism (...) between a system and its environment, which is necessarily imposed by a scientist focusing on the former. The second pertains to the complexity of complex systems; a complex system understood as a system in which different kinds of causal entailments intertwine together. As I discuss in the paper besides developing the argument, perspectivism thus understood ties together several issues ranging from organicism to emergentism and to processual philosophy, and from the ceteris paribus talk of biology to the measurement problem of quantum mechanics. I also discuss Rosen’s relational formalisms as a concrete example of how perspectival epistemology might directly suggest novel strategies and practices of doing theoretical science. (shrink)

With his book The Nature of Mathematical Knowledge (1983), Ph. Kitcher, that had been doing extensive research in the history of the subject and in the contemporary debates on epistemology, saw clearly the need for a change in philosophy of mathematics. His goal was to replace the dominant, apriorist philosophy of mathematics with an empiricist philosophy. The current philosophies of mathematics all appeared, according to his analysis, not to fit well with how mathematicians actually do mathematics. A (...) shift in orientation should invoke the more general reflection that causal, genetic factors are as significant for epistemology as logical structure. As I am to a large extent sympathetic with Kitcher's proposal, my aims here will be simple: first I start presenting Kitcher's argument, and then I try to raise some doubts about his contribution. These doubts come probably from my unskilfulness to follow the torrential flow of Kitcher's ideas. (shrink)

The nature of mathematical knowledge can be understood only by locating the knowing mathematician in an epistemic community. This claim is defended by extending Kripke's version of the Private Language Argument to include informal rules and using Godelian results to argue that such rules rules necessary in mathematics. A committed formalist might evade Kripke's original argument by positing internal mechanisms that determine rule -governed behavior. However, in the presence of informal rules, the formalist position collapses into the (...) extreme skepticism that the Private Language Argument works against. The existence of a community of rule followers provides the only viable alternative to such skepticism. (shrink)

The argument I present here is an example of the manner in which naturalizing epistemology can help address fairly traditional epistemological issues. I develop one argument against coherentist epistemologies of empirical knowledge. In doing so, I draw on BonJour (1985), for that account seems to me to indicate the direction in which any plausible coherentist account would need to be developed, at least insofar as such accounts are to conceive of justification in terms of an agent (minimally) (...) possessing articul able reasons and arguments, as is standard. I end by indicating important elements of coherentist epistemology that can be salvaged in the face of my argument, provided we are willing to drop the traditional commitment to characterizing justification in terms of the structure of articulable argument. (shrink)

Several of Beller, Bender, and Medin’s (2012) issues are as relevant within cognitive science as between it and anthropology. Knowledge-rich human mental processes impose hermeneutic tasks, both on subjects and researchers. Psychology's current philosophy of science is ill suited to analyzing these: Its demand for ‘‘stimulus control’’ needs to give way to ‘‘negotiation of mutual interpretation.’’ Cognitive science has ways to address these issues, as does anthropology. An example from my own work is about how defeasible logics (...) are mathematical models of some aspects of simple hermeneutic processes. They explain processing relative to databases of knowledge and belief—that is, content. A specific example is syllogistic reasoning, which raises issues of experimenters’ interpretations of subjects’ reasoning. Science, especially since the advent of understandings of computation, does not have to be reductive. How does this approach transfer onto anthropological topics? Recent cognitive science approaches to anthropological topics have taken a reductive stance in terms of modules. We end with some speculations about a different cognitive approach to, for example, religion. (shrink)

The article presents philosophical views on music in the context of the transformations of the worldview from Antiquity to the Modern Time. In this research author also mentions the contemporary issues, and uses her own philosophical concept of the music, which can be described as following: the value of music as a substance and the way of the valuable interaction of a person with the world affirm the essence of musical being, in which the invariable principle of Harmony, the (...) principle of Chaos-Form movement, is preserved. Music expresses the fluid essence of the world and changes of being in space and time. Philosophy of music as a field of philosophical knowledge considers music from ontological-epistemological and phenomenological-axiological prospectives, as something more than just a form of art. It explores the deep, ultimate foundations of the existence of music as such and the philosophical and aesthetic foundations of musical art. Since ancient times music has been a representation of the world in the human conscience and served as the harmonic equivalent of cosmological philosophy, mathematics, astronomy, and astrophysics. The scientific view on music was enriched in the Modern Time by the expanded view on the cosmo-sound space reflected in musical art, which at the same time transforms the mathematical ideas of geometricity, squareness etc. The tendency to create integral world music in the musical practice of the XX-XXI centuries explains the attempt of mankind to present music by modern methods of composing musical art as an expression of fear towards the secret Harmony of universal existence, and, on the other hand - as a form of search for salvation and mental balance, intuitively reflecting what is happening in natural science, which more and more points to the abyss of unknowable universe, and the unstable place of man in the world. (shrink)

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in (...) the last century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point (...) of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

Ontology, the ideas we have about the nature of reality, and epistemology, our concepts about how to gain knowledge about the world, are interdependent. Currently, the dominant ontology in science is a materialist model, and associated with it an empiricist epistemology. Historically speaking, there was a more comprehensive notion at the cradle of modern science in the middle ages. Then “experience” meant both inner, or first person, and outer, or third person, experience. With the historical development, experience has (...) come to mean only sense experience of outer reality. This has become associated with the ontology that matter is the most important substance in the universe, everything else—consciousness, mind, values, etc., —being derived thereof or reducible to it. This ontology is insufficient to explain the phenomena we are living with—consciousness, as a precondition of this idea, or anomalous cognitions. These have a robust empirical grounding, although we do not understand them sufficiently. The phenomenology, though, demands some sort of non-local model of the world and one in which consciousness is not derivative of, but coprimary with matter. I propose such a complementarist dual aspect model of consciousness and brain, or mind and matter. This then also entails a different epistemology. For if consciousness is coprimary with matter, then we can also use a deeper exploration of consciousness as happens in contemplative practice to reach an understanding of the deep structure of the world, for instance in mathematical or theoretical intuition, and perhaps also in other areas such as in ethics. This would entail a kind of contemplative science that would also complement our current experiential mode that is exclusively directed to the outside aspect of our world. Such an epistemology might help us with various issues, such as good theoretical and other intuitions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hume’s Natural Philosophy and Philosophy of Physical Science by Matias SlavovKrisztián PeteMatias Slavov. Hume’s Natural Philosophy and Philosophy of Physical Science. London: Bloomsbury Academic, 2020. Pp. 216. Hardcover. ISBN 9781350087866, £95.Although the relationship between Hume and Newton is a recurring theme in the Hume literature, Matias Slavov’s book does not seek to contribute to the debate between the traditional (Hume imitated Newton’s natural philosophy) and the critical (Hume (...) intended his “science of Man” as the foundation of all other sciences) approaches regarding the nature of this relation. The book is intended to be a summary of Hume’s natural philosophy—or rather an aggregation of his comments on natural philosophical topics—pursued by exploring and analysing themes and issues that were prominent in the natural philosophy of the early modern period. In this respect it is a pioneering undertaking.Of course, Slavov does not claim in his book that Hume is a natural philosopher. “[F]irst and foremost... [h]e is not; for his main objective is to establish a new science of human nature” (1). But Slavov does claim that Hume’s natural philosophical views can be understood and valued in themselves and in the philosophical-scientific context of his time as well. The book is not a discussion of the natural philosophical dimensions of the “science of human nature,” but rather, of the natural philosophical views which are reconcilable with his main objective. In this respect, the book is not a methodological approach to Hume’s supposed natural philosophy, but a synthesis of certain elements of Hume’s philosophy that can feature in a consistent natural philosophy. Although his arguments are generally indirect and are more about what Hume seems to be committed to and what follows from his epistemological position rather than his actual positions, Slavov’s conclusions seem mostly convincing.Thus, Slavov’s point is not that the empiricist method is compatible in every detail with Newtonian mechanics, but rather that Hume seems to be committed more to a Cartesian natural philosophy. Slavov does not write about the possibility and significance of applying natural philosophical methods to moral philosophy; rather, he wants to build a complete philosophy of nature around some of Hume’s basic ideas to “fill the gap for a book on Hume’s relation to natural philosophy and philosophy of physical science” (2).The book has two “equally important aims” (ix): to shed more light onto Hume’s relationship to natural philosophy, and to demonstrate that physics and philosophy have overlapping domains. Hume was hardly concerned with physics, so Slavov’s strategy of defining natural philosophy as an overlap between physics and philosophy (chapter 1) does, to some extent, explain the lack of textual evidence in this area. The definition of natural philosophy chosen by Slavov is also intended to ensure [End Page 170] that these two goals are intertwined, since according to Slavov’s definition, natural philosophy is “a grey area between philosophy and physics” (12).Personally, I do not feel comfortable with this definition because it is not informative enough; it says that natural philosophy is more than modern physics (mathematized natural science), but it is not quite philosophy. While the first implication is very agreeable, and a number of scholars have addressed the issue, to assess the second implication, we would need to know exactly what philosophy is in this context. Perhaps the best candidate is metaphysics, even if Slavov keeps the reader somewhat in the dark on this point, despite the fact that chapter 2 is intended to establish precisely that point. Yet, the metaphysics of this period was usually (with the possible exception of Spinoza) used to support natural scientific explanations. I believe that natural philosophy is more than a “grey area” (9, 12, 22), or a contact zone; it is a complex enterprise that evolved in different forms with different emphases: Cartesian mechanics still sought to understand nature, pursuing the Aristotelian ideal of knowledge, albeit rejecting Aristotelian methods, while empiricists increasingly envisaged a more instrumentalist role for natural explanations. Newton himself was caught between the two.In this context it is intriguing whether Hume carries forward Berkeley... (shrink)

In this dissertation I present a new solution to the renowned Gettier problem. My solution, which in a sense represents a defense of a rather traditional epistemological approach, is based upon a distinction between primary and secondary beliefs. I argue that primary beliefs are known if justified and true, whereas secondary beliefs are known if they are believed on the basis of a known primary belief. Much emphasis is put upon defending this approach against potential objections, but I also (...) draw some epistemological and semantical consequences pertaining to such issues as the nature of epistemological justification, the Lottery Paradox, philosophical skepticism and the semantics of belief contexts. (shrink)

This book collects original essays on the epistemology of modality and related issues in modal metaphysics and philosophical methodology. The contributors utilize both the newer metaphysics-first and the more traditional epistemology-first approaches to these issues. The chapters on modal epistemology mostly focus on the problem of how we can gain knowledge of possibilities, which have never been actualized, or necessities which are not provable either by logico-mathematical reasoning or by linguistic competence alone. These issues are (...) closely related to some of the central issues in philosophical methodology, notably: to what extent is the armchair methodology of philosophy a reliable guide for the formation of beliefs about what is possible and necessary. This question also relates to the nature of thought experiments that are extensively used in science and philosophy. Epistemology of Modality and Philosophical Methodology will be of interest to researchers and advanced students working on the epistemology and metaphysics of modality, as well as those whose work is concerned with philosophical methodology more generally. (shrink)

Abstract:Latin American modernity/coloniality studies emerged in the early 1990s from a network of scholars focused on charting the nature and consequences of causal connections between the first appearances of modernity in Europe and Spanish and Portuguese colonialism in the Americas beginning in 1492. In this article, I address primarily epistemological and ontological issues raised by this literature for issues pertaining to the history and philosophy of science. The first section briefly summarizes the sixteenth century differences (...) that were the starting point of the modernity/coloniality analyses and points to just a few of the still-emerging feminist issues about this early era. The second section focuses on the distinctive epistemic and political stances that Latin American decolonial theorists use today to frame such analyses: they intend to do theory “otherwise.” This Latin American project has been conceptualized through Gloria Anzaldúa's influential “borderlands” thinking and aligns with the feminist standpoint methodology and other knowledge-from-below projects that have appeared in all of the social justice movements. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Galen’s Epistemology: Experience, Reason, and Method in Ancient Medicine ed. by R. J. Hankinson and Matyáš HavrdaPatricia MarechalR. J. Hankinson and Matyáš Havrda, editors. Galen’s Epistemology: Experience, Reason, and Method in Ancient Medicine. Cambridge: Cambridge University Press, 2022. Pp. 348. Hardback, $99.99; paperback, $36.99.Galen famously says that some things are “securely known” (bebaiôs gnôston) (De optima doctrina I 41.15 Kühn). But how do we know things, and what (...) entitles us to take our beliefs to be securely grounded? An optimist on this point, Galen is not seduced by hyperbolic skeptical concerns and argues that, if we follow a systematic method of inquiry, we will build theories that are not only persuasive and useful, but also true. Unsurprisingly, Galen’s views on knowledge in general and its foundation are deeply informed by debates about the possibility and nature of medical knowledge. His method integrates both rationalist and empiricist insights, thereby overcoming the divide between two of the main medical sects of his time. According to Galen, reason and experience must be combined in medical theory and practice.This much is well known. The eleven essays in this volume take further steps to understand Galen’s scientific method by exploring in detail how reason and experience contribute to scientific discoveries and demonstrations. With the exception of two essays that tackle questions of general epistemological significance—Galen’s take on skepticism (Barnes) and his theory of vision (Ierodiakonou)—the majority of the contributions offer detailed descriptions of how Galen’s method integrates rational tools with experience. Some of the essays offer a close reading of Galen’s methodological discussions on these issues (Kupreeva, Hankinson, Tieleman), while others analyze his method in action by examining concrete cases and applications (Havrda, Singer, Lewis). As is to be expected, most of the essays focus on knowledge in the domain of natural philosophy—specifically, medicine—but one fruitfully extends the discussion to Galen’s ethical writings (Kaufman). The last two essays address the reception of Galen’s scientific methodology in Alexandrian Late Antiquity (Wakelnig) and the Islamic Middle Ages (Koetschet).This is an excellent volume that deepens our efforts to reconstruct Galen’s views on the foundation of epistēmē. One thread stands out. Galen argues that experience (peira) is indispensable to knowing and understanding the natural world. He also claims that experience that is not guided by reason will not take us far in our inquiries. The contributions to the volume allow us to appreciate two further facts about Galen’s views on the nature of experience. On the one hand, for Galen, experience itself is always cognitively and theoretically loaded since it is categorized and filtered through conceptual and linguistic categories. On the other hand, Galen is aware that sensory experiences are fine-grained, and he suggests that fully capturing the content of experience linguistically is hard and, sometimes, impossible. Scientific statements are often only a rough indication of the complex realities observed and felt by doctors and patients. Thus, errors may arise in our choice of terms, in the statement of scientific propositions, and in our classifications of experience. These two points could be in tension, and readers are left with questions pertaining to how both of them feature in Galen’s account of the content of experience. Regardless, the discussions in this volume reveal Galen’s acute sensitivity to issues related to the structure, cognitive penetration, and theory-ladenness of experience, as well as the problematic relationship between sense data and the basic empirical statements in [End Page 657] our scientific theories. These essays, then, further confirm Galen’s sophistication as a philosopher and scientist, and open further avenues for research on these questions.Since I am not able to comment on all the essays in the detail they deserve, I instead offer a brief indication of the contents of each. I hope this will allow the reader to get a sense of the richness and importance of this collection.In chapter 1, Jonathan Barnes addresses Galen’s dismissal of skepticism. While giving due consideration to his opposition to skepticism, Barnes persuasively shows that Galen fails to put skeptical worries... (shrink)

This book explains exactly what human knowledge is. The key concepts in this book are structures and algorithms, i.e., what the readers “see” and how they make use of what they see. Thus in comparison with some other books on the philosophy of science, which employ a syntactic approach, the author’s approach is model theoretic or structural. Properly understood, it extends the current art and science of mathematical modeling to all fields of knowledge. The link between structure (...) and algorithms is mathematics. But viewing “mathematics” as such a link is not exactly what readers most likely learned in school; thus, the task of this book is to explain what “mathematics” should actually mean. Chapter 1, an introductory essay, presents a general analysis of structures, algorithms and how they are to be linked. Several examples from the natural and social sciences, and from the history of knowledge, are provided in Chapters 2–6. In turn, Chapters 7 and 8 extend the analysis to include language and the mind. Structures are what the readers see. And, as abstract cultural objects, they can almost always be seen in many different ways. But certain structures, such as natural numbers and the basic theory of grammar, seem to have an absolute character. Any theory of knowledge grounded in human culture must explain how this is possible. The author’s analysis of this cultural invariance, combining insights from evolutionary theory and neuroscience, is presented in the book’s closing chapter. The book will be of interest to researchers, students and those outside academia who seek a deeper understanding of knowledge in our present-day society. (shrink)

This book provides a survey of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the (...) papers in this volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike. (shrink)

One of the chief concerns of the young Descartes was with what he, and others, termed “physico-mathematics”. This signalled a questioning of the Scholastic Aristotelian view of the mixed mathematical sciences as subordinate to natural philosophy, non explanatory, and merely instrumental. Somehow, the mixed mathematical disciplines were now to become intimately related to natural philosophical issues of matter and cause. That is, they were to become more ’physicalised’, more closely intertwined with natural philosophising, regardless of which species (...) of natural philosophy one advocated. A curious, short-lived yet portentous epistemological conceit lay at the core of Descartes’ physico-mathematics—the belief that solid geometrical results in the mixed mathematical sciences literally offered windows into the realm of natural philosophical causation—that in such cases one could literally “see the causes”. Optics took pride of place within Descartes’ physico-mathematics project, because he believed it offered unique possibilities for the successful vision of causes. This paper traces Descartes’ early physico-mathematical program in optics, its origins, pitfalls and its successes, which were crucial in providing Descartes resources for his later work in systematic natural philosophy. It explores how Descartes exploited his discovery of the law of refraction of light—an achievement well within the bounds of traditional mixed mathematical optics—in order to derive—in the manner of physico-mathematics—causal knowledge about light, and indeed insight about the principles of a “dynamics” that would provide the laws of corpuscular motion and tendency to motion in his natural philosophical system. (shrink)

In spite of Ernst Cassirer’s criticisms of psychologism throughout Substance and Function, in the final chapter he issues a demand for a “psychology of relations” that can do justice to the subjective dimensions of mathematics and natural science. Although these remarks remain somewhat promissory, the fact that this is how Cassirer chooses to conclude Substance and Function recommends it as a topic worthy of serious consideration. In this paper, I argue that in order to work out the details of (...) Cassirer’s psychology of relations in Substance and Function, we need to situate it within two broader frameworks. First, I position Cassirer’s view in relation to the view of psychology and logic endorsed by his Marburg Neo-Kantian predecessors, Hermann Cohen and Paul Natorp. Second, I augment Cassirer’s early account of the psychology of relations in Substance and Function with the more mature view of psychology that he presents in The Philosophy of Symbolic Forms. By placing Cassirer’s account within these contexts, I claim that we gain insight into his psychology of relations, not just as it pertains to mathematics and natural science, but to culture as a whole. Moreover, I maintain that pursuing this strategy helps shed light on one of the most controversial features of his philosophy of mathematics and natural science, viz., his theory of the a priori. (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and philosophy (...) of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)

If we want to understand why mathematical knowledge is extraordinarily reliable, we need to consider both the nature of mathematical arguments and mathematical practice as a social practice. Mathematical knowledge is extraordinarily reliable because arguments in mathematics take the form of deductive mathematical proofs. Deductive mathematical proofs are surveyable in the sense that they can be checked step by step by different experts, and a purported proof is only accepted as a (...) proof by the mathematical community once a number of experts have checked the proof. Hence, the reliability of the body of mathematical knowledge is in part obtained through the surveyability of proofs and the social process of proof validation. This chapter reviews work relating to (1) the norm in the mathematical community of only counting as argument deductive mathematical proof, and (2) the norm of only counting as deductive mathematical proof an argument that has been confirmed to be a deductive mathematical proof by a number of experts. The chapter also presents cases of unusual mathematical proofs or arguments that challenge these norms. (shrink)

This article is devoted to the philosophical study of the conditions under which knowledge can become a component or tool of education. The presentation of the contribution of epistemology to human development and education is based on addressing issues such as the nature of knowledge, sources of knowledge, theories, and criteria of truth. We proceed from the idea that knowledge is a condition of education. Particular attention is paid to the issue of distinguishing between (...) such types of knowledge as 'knowing how’ and ‘knowing that'. Educational practices open a common foundation that unites the life world of people, types of sociality, and hermeneutical practices. The epistemological approach to the question of the essence of knowledge assumes that knowledge meets three requirements, namely, knowledge must be objective, subjective, and evidentiary. Epistemology includes subjectivity as the basis of human existence in the natural world and the world with others. Intersubjectivity is considered a criterion for the reliability of knowledge about the world, which allows asserting the relationship of the objective with the subjective. (shrink)

The ArgumentMany of the epistemological issues that occupied natural philosophers of the seventeenth century were expressed visually in title-page engravings. One of those issues concerned the relative status to be accorded to evidence of the senses, as compared to knowledge gained by faith or reason. In title-page illustrations, the various arguments were often waged by a series of light metaphors: the Light of Reason, the Light of Nature, and the Lights of Sense, Scripture, and Grace. (...) When such illustrations are examined with the authors' theological views in mind, it becomes apparent that in the first half of the seventeenth century, Catholic authors favored the Light of Reason as a source of truth, while Protestant authors favored the Light of Nature. Since by the end of the century it was widely accepted by scientists of all religious persuasions that certain knowledge must be grounded in sense evidence and the direct study of nature, one might argue that in this instance Protestantism was responsible for nurturing an important development of the Scientific Revolution. However, the skewed nature of the sample and the large number of counterexamples available mitigate against taking this offshoot of the Merton thesis too seriously. (shrink)

This book aims to provide scientists and engineers, and those interested in scientific issues, with a concise account of how the nature of scientific knowledge evolved from antiquity to a seemingly final form in the Twentieth Century that now strongly limits the knowledge that people would like to gain in the Twenty-first Century. Some might think that such issues are only of interest to specialists in epistemology (the theory of knowledge); however, today's major scientific (...) and engineering problems--in biology, medicine, environmental science, etc.--involve enormous complexity, and it is precisely this complexity that runs up against the limits of what is scientifically knowable. To understand the issue, one must appreciate the radical break with antiquity that occurred with the birth of modern science in the Seventeenth Century, the problems of knowledge and truth engendered by modern science, and the evolution of scientific thinking through the Twentieth Century. While originally aimed at practicing scientists and engineers, it is my hope that this book can provide a generally educated person with a basic understanding of how our perspective on scientific knowledge has evolved over the centuries to escape pre-Galilean commonsense thinking. Such an appreciation is not only beneficial for one's general education, but is important for non-scientists who must teach young students or make policy decisions in government or business. (shrink)

This book, first published in 1975, is an examination of the theoretical foundation of the sociology of art and literature and an in-depth study in the sociology of knowledge. In discussing and clarifying some of the important philosophical issues in this field, the constant underlying reference is to the creative and artistic-expressive areas of knowledge ¿ so that the better understanding of the social nature and genesis of all knowledge may point the way towards a (...) similar comprehension of art and society. (shrink)

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

The great mathematician, physicist, and philosopher, Hermann Weyl, once called mathematics the “science of the infinite.” This is a fitting title: contemporary mathematics—especially Cantorian set theory—provides us with marvelous ways of taming and clarifying the infinite. Nonetheless, I believe that the epistemic significance of mathematical infinity remains poorly understood. This dissertation investigates the role of the infinite in three diverse areas of study: number theory, cosmology, and probability theory. A discovery that emerges from my work is that the epistemic (...) role of the infinite varies, often in surprising ways, across different domains of knowledge. -/- My first chapter examines the role of mathematical infinity in number theory. It is reasonable to think that theorems concerning finite patterns and structures in the natural numbers are particularly “simple” or “elementary.” Indeed, such statements are comprehensible to a wide range of investigators, regardless of their mathematical training. One might then expect proofs of these theorems to be similarly comprehensible. However, many proofs, especially those that utilize only finitary methods, are exceedingly difficult to understand. Consequently, one finds that finitary theorems are often re-proved using infinitary techniques. My claim is that this is because infinitary proofs are often explanatory, while finitary proofs are not. This chapter analyzes why this is the case. Along the way, I investigate other questions of long-standing interest in the philosophy of mathematics, e.g., the role of purity/impurity ascriptions and the nature of the content of a theorem. In particular, I diagnose the explanatory potential of the infinite by articulating a new construal of content. This new construal both saves intuitive epistemic ascriptions made in mathematical practice and explains the unexpected role of the infinite in providing explanatory proofs of finitary statements. Thus, in number theory, my claim is that the infinite often plays an explanatory role. -/- My second chapter turns to the role of the infinite in cosmology. It investigates a question much discussed by philosophers and physicists alike: is the spatial extent of the universe finite or infinite? Contemporary cosmological research has indicated that one of the essential determinants of the extent of the universe is the topology we ascribe to space. Topology is a global property, which may suggest that it is not testable through local observation. Nonetheless, some cosmologists have indicated that it may be empirically detected, thereby providing an answer to the question of spatial extent. I argue that, in fact, the epistemic status of the topology of space is extremely subtle and not well captured by any of the categories commonly employed by philosophers of science. In particular, I argue that topological properties are neither empirical nor a priori (even in suitably weakened senses). Furthermore, I claim that we should prefer topological properties that generate finite universe models (consistent with our best data) in order to avoid extremely thorny issues concerning the physics of an infinite universe. I argue for such a preference on the grounds of the simplicity and explanatory power of finite universe models. Thus, in cosmology, my claim is that the finite often plays an explanatory and simplifying role. -/- My third chapter investigates several paradoxes that arise in the foundations of infinitary probability theory: the Label Invariance Paradox, God’s Lottery, and Bertrand’s Paradox. I argue that these have been poorly understood because they do not expressly concern probability theory, but rather our intuitions about—and formal techniques for dealing with— infinite sets. The paradoxes in question are, in fact, symptoms of our complete reliance upon Cantorian cardinality and its associated criterion of sameness of “size.” That is, two sets have the same cardinality if and only if the elements of the sets can be placed in 1-1 correspondence. When applied to infinite sets, this criterion produces counterintuitive verdicts. For instance, given a fair lottery on the natural numbers, we expect that the probability of drawing an even number is 1/2, and likewise for drawing an odd number. However, one can construct “relabellings” of the naturals such that the probability of drawing an even number remains 1/2, while the probability for drawing an odd number becomes 1/4. I argue that, ultimately, it is the coarseness of Cantorian cardinality that generates the probabilistic paradoxes. I then propose that finer-grained measures of infinite sets from mathematical logic and number theory can help to dissolve the paradoxes in question. Thus, in probability theory, we find that particular kinds of infinitary techniques effectively systematize our theory, while others lead to paradox. (shrink)

Can epistemic logic state anything interesting about the epistemology of mathematics? That's one of Jaakko Hintikka’s claims. Hintikka was not only the founder of modal epistemic logic (1962), since he also worked on the foundations of mathematics (1996). Using what he calls "second generation" epistemic logic (2003), i.e. independence-friendly (IF) epistemic logic, Hintikka revisits the epistemology of mathematics, and in particular the debate between classical and intuitionistic mathematics (2001). The aim of the talk is to show that Hintikka is right (...) regarding IF epistemic logic, for such a logic enables us to account for interesting features of mathematical knowledge. However, the path is not as easy as that Hintikka suggests. I will show that the well-known issue of logical omniscience directly threatens the understanding of intuitionism offered by IF epistemic logic. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, (...) that numbers are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

This book interrogates the ontology of mathematical entities in Spinoza as a basis for addressing a wide range of interpretive issues in Spinoza’s epistemology—from his antiskepticism and philosophy of science to the nature and scope of reason and intuitive knowledge and the intellectual love of God. Going against recent trends in Spinoza scholarship, and drawing on various sources, including Spinoza’s engagements with optical theory and physics, Matthew Homan argues for a realist interpretation of geometrical figures in (...) Spinoza; illustrates their role in a Spinozan hypothetico-deductive scientific method; and develops Spinoza’s mathematical examples to better illuminate the three kinds of knowledge. The result is a portrait of Spinoza’s epistemology as sanguine and distinctive yet at home in the new Cartesian and Galilean scientific-philosophical paradigm. (shrink)

The Nature of Selection is a straightforward, self-contained introduction to philosophical and biological problems in evolutionary theory. It presents a powerful analysis of the evolutionary concepts of natural selection, fitness, and adaptation and clarifies controversial issues concerning altruism, group selection, and the idea that organisms are survival machines built for the good of the genes that inhabit them. "Sober's is the answering philosophical voice, the voice of a first-rate philosopher and a knowledgeable student of contemporary evolutionary theory. His (...) book merits broad attention among both communities. It should also inspire others to continue the conversation."-Philip Kitcher, Nature "Elliott Sober has made extraordinarily important contributions to our understanding of biological problems in evolutionary biology and causality. The Nature of Selection is a major contribution to understanding epistemological problems in evolutionary theory. I predict that it will have a long lasting place in the literature."-Richard C. Lewontin. (shrink)

In his Contra Academicos, Augustine offers one of the most detailed responses to scepticism to have come down to us from antiquity. In this paper, I examine Augustine’s defence of the existence of infallible knowledge in Contra Academicos 3. I challenge a number of established views (including those of Myles Burnyeat, Gareth Matthews, and Christopher Kirwan) concerning the nature and merit of Augustine’s defence of knowledge and propose a new understanding of Augustine’s response to scepticism (including his (...) semantic response to external world scepticism) and several important elements of Augustine’s thought concerning signification, cognition, and object-directed thought. I argue that once we properly understand Augustine’s views about these issues, his arguments in defence of knowledge are more interesting and more successful than usually thought. (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)

This handbook provides a comprehensive overview of Islamic ethical issues within a wide spectrum of philosophy of science topics, examining the development of the model of moral inclusiveness in economics, science and society from ontological, epistemological and analytical perspectives. This paradigm takes the view that ethics is systemically endogenous, and can be studied by the most rigorous scientific analysis pertaining to diverse issues and problems of ethicality in socio-scientific inquiry. This handbook takes a sweeping transdisciplinary approach (...) that is deeply phenomenological, to the nature and logic of scientific inquiry of, and in, the Qur’an. Such an approach invokes the episteme of the unity of knowledge in the socio-scientific systemic sense. The volumes' respective sections focus on the nature, logic, and role that ethics plays in formulating new vistas of alternative epistemic futures in Islamic economics, finance, and the social sciences. The ideas presented are situated within the broader context of the post-modernist, post-pandemic, and the post-Covid-19 epoch, while being aimed at conceptualising a distinctive new outlook of transdisciplinary intellection in Islamic philosophy. The content is rigorously conceptual, qualitative, quantitative, and applied. Covering a diversity of subject areas from philosophy of science, to economics and in socio-scientific context within the realm of Islamic philosophy, this forms a key text for scholars in these respective arenas, led by pioneering scholarship in Islamic studies. (shrink)

Arthur Kaufmann is one of the most prominent figures among the contemporary philosophers of law in German speaking countries. For many years he was a director of the Institute of Philosophy of Law and Computer Sciences for Law at the University in Munich. Presently, he is a retired professor of this university. Rare in the contemporary legal thought, Arthur Kaufmann's philosophy of law is one with the highest ambitions — it aspires to pinpoint the ultimate foundations of law by explicitly (...) proposing an ontology, a general theory of knowledge and concept of a person. Kaufmann's work derives, first of all, from the thinking of Gustav Radburch, his teacher, and then from ideas of Karl Engish and Hans-Georg Gadamer. The philosophy undertakes to pursue the ultimate foundation of law, law which is understood by Kaufmann, first of all, as a "concrete judgement" that is, what is right in a concrete situation. Justice belongs to the essence of law and "unjust law" is contradictio in adiectio. Kaufmann opposes all those theories, which as the only foundation for establishing just law (Recht) adopt legal norms (Gesetz). In Kaufmann's opinion , such theories are powerless in the face of all types of distortions of law rendered by political forces. He suggests that the basic phenomenon which needs to be explained and which cannot be disregarded by a philosopher of law is so-called "legal lawlessness" ("Gestzliches Unrecht"). "Legal lawlessness" which forms a part of life experience for the people of twentieth century totalitarian states. It proved "with the accuracy of scientific experiment" that the reality of law consists of something more than bare conformity with legal norms. The existence of lex corrupta indicates that law contains something "non-dispositive" which requires acknowledgment of both law-maker and judge. Kaufmann, accepting the convergent concept of truth and cognition, assumes that "non-dispositive" content, emerging as the conformity of a number of cognitive acts of different subjects (inter-subjective communicativeness and verifiability), indicates the presence of being in this cognition. The questions "What is law?" and "What are the principles of a just solution?" lead straight to the ontology of law, to the question about the ontological foundations of law. Kaufmann discerns the ontological foundations of law in the specifically understood "nature of things" and, ultimately, in a "person". He proposes a procedural theory of justice, founded on a "person". In my work, I undertake to reconstruct the train of thought which led Kaufmann to the recognition of a "person" as the ontological foundation of law. In the first part, the conception of philosophy adopted by Kaufmann, initial characteristics of law — of reality which is the subject of analysis, as well as, the requirements for proper philosophical explanation of law posed by Kaufmann are introduced. In the second, Kaufmann's reconstruction of the process of the realisation of law is presented. Next, the conception of analogy which Kaufmann uses when explaining law is analyzed. In the fourth part, Kaufmann's conception of ontological foundations of law is discussed. A critical analysis is carried out in which I demonstrate that the theory of the ontological foundation of law proposed by Kaufmann and the concept of a person included in it do not allow a satisfac¬tory explanation of the phenomenon of "legal lawlessness" and lead to a number of difficulties in the philosophical explanation of law. Finally, the perspectives of a proper formulation of the issue of the ontological foundations of law are drafted in the context of the analyzed theory. My interest is centered on the conception of philosophy adopted by Kaufmann, according to which the existence of the reality is inferred on the basis of a certain configuration of the content of consciousness, whereas at the point of departure of philosophy of law, the data to be explained is a certain process, which is, basically, a process of cognition, while the reality appears only as a condition for the possibility of the occurrence of the process. I wish to argue that the difficulties which appear in the explanation of law are a consequence of the assumed fundamental philosophical solu¬tions, which seem to be characteristic, though usually not assumed explicitly, in philoso¬phy and theory of law dominant at present in continental Europe. Thereby, I wish to show the significance of ontological and epistemological solutions to the possibility of a proper formulation of the problems posed by philosophy and theory of law. Kaufmann proclaims himself in favour of a philosophy which poses questions about the ultimate foundations of understanding of the reality. In epistemology, he assumes that answers to the questions "What is reality like?" and ultimately "What is real?" are inferred on the basis of uniformity of a cognitive acts of different subjects. Cognition of the reality is accomplished exclusively through the content of conceptual material. The two fundamental questions posed by philosophy of law are "What is just law?" and "How is the just law enacted?" The latter is a question about the process of achieving a solution to a concrete case. Since, in Kaufmann's opinion, law does not exist apart from the process of its realisation, an answer to the question about the manner of realisation of law is of fundamental significance to answering the question: "What is law?" and to the explanation of the question about the ontological grounding of law, which is, as well, the foundation of justice. The proper solution has to take into account the moment of "non-dispositive" content of law; its positiveness understood as the reality and, at the same time, it has to point to the principles of the historical transformation of the content. Law, in the primary meaning of the word, always pertains, in Kaufmann's opinion, to a concrete case. A legal norm is solely the "possibility" of law and the entirely real law is ipsa res iusta, that which is just in a given situation. Determination of what is just takes place in a certain type of process performed by a judge (or by man confronted with a choice). Kaufmann aims to reconstruct this process. A question about the ontological foundation of law is a question about the ontological foundations of this process. In the analyzed theory it is formulated as a question about the transcendental conditions, necessary for the possibility of the occurrence of the process: how the reality should be thought to make possible the reconstructed process of the realisation of law. Kaufmann rejects the model for finding a concrete solution based on simple subsump¬tion and proposes a model in which concrete law ensues, based on inference by analogy, through the process of "bringing to conformity" that which is normative with that which is factual. Kaufmann distinguishes three levels in the process of the realisation of law. On the highest level, there are the fundamental legal principles, on the second legal norms, on the third — concrete solutions. The fundamental principles of law are general inasmuch as they cannot be "applied" directly to concrete conditions of life, however, they play an important part in establishing norms. A judge encounters a concrete situation and a system of legal norms. A life situation and norms are situated on inherently different levels of factuality and normativeness. In order to acquire a definite law both a norm (system of norms) and a life situation (Lebenssachverhalt) should undergo a kind of "treatment" which would allow a mutual conformity to be brought to them. Legal norms and definite conditions of life come together in the process of analogical inference in which the "factual state" ("Tatbestand") — which represents a norm, and in the "state of things" ("Sachverhalt") — which represents a specific situation are constructed. A "factual state" is a sense interpreted from a norm with respect to specific conditions of life. The "state of things" is a sense constructed on the basis of concrete conditions of life with respect to norms (system of norms). Legal norms and concrete conditions of life meet in one common sense established during the process of realisation of law. Mutatis mutandis the same refers to the process of composition of legal norms: as the acquisition of concrete law consists in a mutual "synchronization" of norms and concrete conditions of life, so acquisition of legal norms consists of bringing to conformity fundamental principles and possible conditions of life. According to Kaufmann, both of these processes are based on inference through analogy. As this inference is the heart of these processes it is simultaneously a foundation finding just law and justice. How does Kaufmann understand such an inference? As the basis for all justice he assumes a specifically interpreted distributive justice grounded on proportionality. Equality of relations is required between life conditions and their normative qualification. Concrete conditions of life are ascribed normative qualification not through simple application of a general norm. More likely, when we look for a solution we go from one concrete normative qualified case to another, through already known "applications" of norms to a new "application". The relation between life conditions and their normative qualifica¬tion has to be proportional to other, earlier or possible (thought of) assignments of that which is factual to that which is normative. Law as a whole does not consist of a set of norms, but only of a unity of relations. Since law is a, based on proportion, relative unity of a norm and conditions of life, in order to explain law in philosophical manner, the question about ontological base of this unity has to be asked. What is it that makes the relation between a norm and conditions of life "non-dispositive"? What is the basis for such an interpretation of a norm and case which makes it possible to bring a norm and conditions of life into mutual "conformity"? This is a question about a third thing (next to norms and conditions of life), with respect to which the relative identity between a norm and conditions of life occurs, about the intermediary between that which is normative and that which is factual and which provides for the process of establishing of norms, as well as, finding solutions. It is the "sense" in which the idea of law or legal norm and conditions of life have to be identical to be brought to mutual "conformity". In Kaufmann's opinion such a sense is nothing else but the "nature of things" which determines the normative qualification of the reality. Since establishment of this "sense" appears to be "non-dispositive" and controlled inter-subjectively (namely, other subjects will reach a similar result) so, in conformity with the convergent concept of truth, the "nature of things" must be assigned a certain ontological status. According to Kaufmann this is a real relation which occurs between being and obligation, between the conditions of life and normative quality. However, it should be underlined that from the point of view of the analyzed system the "nature of things" is a correlate of constructed sense, a result of a construction which is based on the principle of consistent understanding of senses ("non-normative" and "normative") and is not a reality which is transcendent against the arrangement of senses. In Kaufmann's theory, inference from analogy appears to be a process of reshaping the concepts (senses) governed by tendency to understand the contents appearing in relations between that which is factual and that which is normative in a consistent way. The analogical structure of language (concepts) and recognition of being as composed of an essence and existence is an indispensable requirement for the possibility of the realisation of law, based on specifically understood inference from analogy. It is necessary to assume a moment of existence without content which ensures unity of cognition. Existence emerges thus as a condition of the possibility of cognition. According to Kaufmann, the "nature of things" is the heart of inference through analogy and the basis for establishment of finding of law. Inference from the "state of things" to a norm or from a norm to the "state of things" always means inference through the "nature of things". The "nature of things" is the proper medium of objective legal sense sought in every cognition of law. In Kaufmann's view, the question whether the "nature of things" is ultima ratio of interpretation of law or is only a means of supplement gaps in law or whether it is one of the sources of law, is posed wrongly. The "nature of things" serves neither to supplement the gaps nor is it a source of law as, for example, a legal norm may be. It is a certain kind of "catalyst" necessary in every act of making law and solving a concrete case. Owing to "nature of things" it is possible to bring to a mutual conformity the idea of law and possible conditions of life or legal norms and concrete conditions of life. In Kaufmann's conception the "nature of things" is not yet the ultimate basis for understanding the "non-dispositiveness" of law. The relation between obligation and being is determined in the process of the realisation of law. Both the process itself and that which is transformed in this process are given. A question about the ontological bases of "material" contents undergoing "treatment" in the process of the realisation of law and about being which is the basis of regularity of the occurrence of the process arises. Only this will allow an explanation that the result of the process is not optional. Thus, a question about reality to which law refers and about the subject realising the law has to be formed. To this, Kaufmann gives the following answer: that which is missing is man but not "empirical man" but man as a "person". A "person" understood as a set of relations between man and other people and things. A "person" is the intermediary between those things which are different — norm and case are brought to conformity. A "person" is that which is given and permanent in the process of the realisation of law. It determines the content of law, is "subject" of law; this aspect is described by Kaufmann as the "what" of the process of realisation of law. A "person" consists of precisely just these relations which undergo "treatment" in the process. On the other hand, a "person" is "a place" in which the processes of realisation of law occur, it is the "how" of normative discourse, a "person" is that which determines the procedure of the process, being "outside" of it. This aspect of a "person" is connected with the formal moment of law. A "person" being, at the same time, the "how" and the "what" of the process of the realisation of law, is also, to put it differently, a structural unity of relation and that which constitutes this relation (unity of relatio and relata). According to this approach a "person" is neither an object nor a subject. It exists only "in between". It is not substance. Law is the relation between being and obligation. That which is obligatory is connected with that which is general. That which is general does not exist on its own, it is not completely real. Accordingly, a "person" as such is also not real. It is relational, dynamic and historical. A "person" is not a state but an event. In Kaufmann's opinion, such a concept of a "person" helps to avoid the difficulties connected with the fungibility of law in classical legal positivism. A "person" is that which is given, which is not at free disposal and secures the moment of "non-dispositiveness" of law. Kaufmann concludes: "The idea (»nature«) of law is either the idea of a personal man or is nothing". Theory points at the structure of realising law and explains the process of adoption of general legal norms for a concrete situation. The analysis has shown however, that in this theory a satisfactory answer to the question about the ultimate foundations of law is not given. It seems that in the analyzed theory the understanding of human being takes place through understanding of law. What is good for man as a "person", what is just, what a "person" deserves may be determined only against the existing system of law. A "per¬son" adopted as a basis of law is the reality postulated in the analysis of the process of the realisation of law. It is a condition of possibility of this process ( explaining, on one hand, its unity and, on the other hand, the non-dispositive moments stated in this process). A "person" in the discussed theory is entirely defined by the structure of law, it can be nothing more than that which is given in law, what law refers to, what law is about. Being, which is a "person", is constituted by relations between people and objects, the relations which are based on fundamental links between norms and conditions of life established in a process of bringing them to conformity. It has to be assumed that man as a "person" is a subject of law only as far as realising law "treats" given senses according to their current configuration. The system of law is a starting point and it describes in content what man is as a "person". Moreover, being a "person" is the condition for entering legal relations. Consistently, Kaufmann writes that "empirical man" is not the subject of law, man is not "out of nature" a "person". People become "persons" due to the fact that they acknowledge each other as "persons" — acknowledging, at the same time, law. This acknowledgement is a con¬dition of existence, of the possibility of the occurrence of process of realisation of law and of constituting legal relations which ultimately constitute a "person". Kaufmann assumes, that law tends towards a moral aim: it may and must create an external freedom, without which the internal freedom to fulfil moral obligations cannot develop. However, this postulate is not based on the necessary structure of human being. From the point of view of his system, it is nothing more than only a condition for the possibility of the occurrence of the process of the realisation of law — lack of freedom would destroy the "how" of this process. Thus, the postulate to protect the freedom of personal acts has to be interpreted, in accordance with the analyzed theory, as a postulate, the fulfilment of which aims ultimately at the accomplishment of the very same process of realisation of law itself and not the realisation of a given man. Kaufmann considers a "person" to be an element which unites the system of law as a whole. Law is a structure of relations, which are interdependent and inter-contingent. Consequently, a "person" which is to form the ontological basis of law has to be entity consisting of all relations. Being also the "how" of the process of realisation of law, if a "person" is to warrant its unity, it has to be a common source for all procedures. Hence, a single "person" would constitute a subject of law. Man appears to be only a moment of a certain entirety, realisation of which should be an aim of his actions. Law, creating a "person" as an object and subject of law becomes a primary entity. In the analyzed theory, the basis for determination of aims which law sets to man is not the allocation of man-subject to something which improves him but rather, such relation is only just constituted by law. A question appears, why should aims set in law also be the aims of "empirical man"? Why is this "empirical man" to be punished in the name of a "person" understood in such a way? If, however, it is assumed that what is man is determined by a system which is superior to him, then man has to be understood only as a part of a whole and there are no grounds to prohibit istrumental treatment of man and so the road to all aspects of totalitarianism might be opened. A problem of the application of created theory to the reality arises, the reality which the theory pretends to explain. Ultimately in his theory Kaufmann does not give any systemic grounds for a radical questioning of the validity of any legal norms. Every new norm becomes an equal part of system of norms. It is only its interpretation and application to given conditions of life that may be disputable, however, this refers to all norms without exception. Cohesive inter-pretation of norms and applications is necessary and sufficient for the acquisition of just law. New norms have to be interpreted in the light of others, correspondingly, the other norms require reinterpretation in the light of the new ones. Contradiction in interpretation of a norm does not form a basis for questioning norms but may serve only to question the manner of their interpretation (understanding). Therefore, no grounds exist to assume any legal norm as criminal or unjust, and in consequence, to question any consistently realised system based on formally, properly established norms, as "legal lawlessness". As law and a "person" do not exist without the process of realisation of law, the role of legal safety becomes crucial as the condition for the possibility of the occurrence of the process of realisation of law. Denying legal safety would be tantamount to negation of law in general (also of moral law) as negation of safety takes away, at the same time, the basis for occurrence of the process of realisation of law. Moreover, any lack of legal safety would also mean lack of a basis for the existence of man as a "person". Kaufmann's thesis, that civil disobedience is legalized only when it has a chance to lead to success, consistent with his concept of the foundations of law, seems to point directly to conclusions which deny the facts taken under consideration and doubtlessly Kaufmann's own intentions, since it would have to be assumed that accordingly there are no grounds to question a legal system in force based on violence which secures its operation. Force finally seems to determine which one of the mutually irreconcilable normative systems constitute law and which does not. A legitimate position is one which leads to success, it is the weaker system which is negated. If so, then basically violent imposition of law is not an act directed against the law in force but, to the contrary, realisation of law. In the context of the new system the former system of law may be talked about as unjust solely in the sense of being incapable of being consistently united with the new. However, at the base, ultimately, lies force which reaffirms differences and excludes from the process of realisation of law certain norms and their interpretations. Kaufmann was aiming at grounding of that which is "non-dispositive" in a certain given framework of interpretation. Nevertheless, he does not provide foundations for the understanding of phenomena, which he undertakes to explain at a point of departure. Instead of explaining them the theory negates the possibility of their existence. The reality postulated in regard to "non-dispositive" moments of the reconstructed process of acquiring law consist of a specifically understood "person", which appears in Kaufmann's conceptions as a condition of the possibility of the realisation of law. According to this approach understanding of a "person" may be only a function of law. To understand "legal lawlessness" and foundations of justice it is necessary to look for such theory of law in which understanding of man as a "person" and being is not a function of understanding of law (in which a "person" is not only a condition for possibility of reconstructed process of realisation of law; for possibility of cognition processes). It seems necessary to start from theory of being and a "person" based on broader experience than the one assumed by Kaufmann and reconstruct the ontological foundations of the process of realisation of law only in such perspective. Kaufmann points out that that to which law refers is ipsa res iusta a concrete relation of man to other people and things. This relation, in his theory, appears to be basically only just constituted by law (normative senses "applied" to conditions of life). Therefore, understanding the relation between a given man and other people and things which constitute the aim of his actions, that is understanding of good, is enacted against the background of constitution of senses; constitution which is a result of a process aiming towards consistent understanding of particular contents (of nor¬mative and non-normative senses). "Being" is secondary towards constructed senses it is only their correlate. The primary relation consists of relation of a man to law (system of norms), while the secondary relation is one of man to something which is the aim of his action (relation between man and good). Considering such approach it is difficult to envision a satisfying answer to the fundamental question: why does law put concrete man under any obligation to obey it? The source of this problem can be seen in reduction of the base for understanding good to content of obligation formulated in auto-reflection. Such reduction seems to be a consequence of Kaufmann's adoption of "convergent concept of truth" and in con¬sequence his recognition of indirect, essentialistic grasp of reality formulated in concepts as the basic and only foundation of theory of being and of law. In view of such an approach, analogy of law, concepts and being is the condition for the possibility of the process of transformation of senses which aims at consistent interpretation of all law. Existence is postulated with respect to the possibility of unity of experience and cognition. However, also a different approach to understanding of the problem of being and good is possible. In spontaneous cognition being is affirmed, first of all, not as a certain, non-contradictory, determined content, but as something existing. Together with a cer¬tain content (passed indirectly through notions) existence of being is co-given. The basis for unity of being is not formed by the consistence of content, as it is in the case of the theories departing from the analysis of cognition processes, but by an act of existence realising content (essence). Such an approach makes it also possible to go beyond the convergent concept of truth. It is worth mentioning that allocation of an agent to good is realised not only by the content of duty. A statement that something is good is primary with respect to determination of this good in content. The recognised good always bears some content, however, there are no reasons to base the concept of good exclusively on indirect, formulated in concepts cognition. As primary, can be adopted the relation of man to good and not of man to law. Determination in content appears to be only an articulation of aspectual cognition of being, as an object of action. In such a case the basis for relative unity of norm and conditions of life is not the "nature of things" understood as correlate of sense but it is relation to good based on internal constitution of man as potential, not self-sufficient being. It does not mean, that the moments of the process of realisation of law singled out by Kaufmann are not important to determination of what is just. He, quite rightly, points to significant role played by norms in the evaluation of concrete situations, in man's search for closer specification in content of good innate to him. The structure of process of determining law for a concrete situation, to a great degree corresponds to the processes of determining law which take place not only in the legal sciences. Kaufmann's analyses of the process of realisation of law show the complexity of the structure of these processes and point towards important moments allowing a better understanding of law and man. Nevertheless, these analyses cannot be a basis for construction of philosophical theory of law, theory which hopes to point out the ultimate, ontological foundations for understanding law. Kaufmann's results may become fully valid only in a more general perspective including broader experience at the point of departure. (shrink)

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

According to a very common view, the main tenet of empiricism is the conviction that all human knowledge derives from sensory experience. But classic philosophers representing empiricism hold that mathematical knowledge is a priori. Mill intended to demonstrate that the laws of arithmetic and geometry have inductive origins. But Frege and others authors showed that Mill’s arguments were wrong. Benacerraf held that, since mathematical objects are abstract entities, they could not have any causal relationship with human (...) beings, so they cannot be known by us. On the other hand, biology and psychology show that in animals and human creatures we can find innate behaviours, in accordance with the theory on natural selection. Experiments performed by Wynn and by other psychologists strongly support that very young babies can determine the results of simple arithmetical operations without any previous learning. We conclude that there are convincing reasons to accept the rationalist thesis about the a priori character of mathematical knowledge. (shrink)

The dispute over the viability of various theories of relativistic, dissipative fluids is analyzed. The focus of the dispute is identified as the question of determining what it means for a theory to be applicable to a given type of physical system under given conditions. The idea of a physical theory's regime of propriety is introduced, in an attempt to clarify the issue, along with the construction of a formal model trying to make the idea precise. This construction involves a (...) novel generalization of the idea of a field on spacetime, as well as a novel method of approximating the solutions to partial-differential equations on relativistic spacetimes in a way that tries to account for the peculiar needs of the interface between the exact structures of mathematical physics and the inexact data of experimental physics in a relativistically invariant way. It is argued, on the basis of these constructions, that the idea of a regime of propriety plays a central role in attempts to understand the semantical relations between theoretical and experimental knowledge of the physical world in general, and in particular in attempts to explain what it may mean to claim that a physical theory models or represents a kind of physical system. This discussion necessitates an examination of the initial-value formulation of the partial-differential equations of mathematical physics, which suggests a natural set of conditions---by no means meant to be canonical or exhaustive---one may require a mathematical structure, in conjunction with a set of physical postulates, satisfy in order to count as a physical theory. Based on the novel approximating methods developed for solving partial-differential equations on a relativistic spacetime by finite-difference methods, a technical result concerning a peculiar form of theoretical under-determination is proved, along with a technical result purporting to demonstrate a necessary condition for the self-consistency of a physical theory. (shrink)

Science education researchers have long advocated the central role of the nature of science for our understanding of scientific literacy. NOS is often interpreted narrowly to refer to a host of epistemological issues associated with the process of science and the limitations of scientific knowledge. Despite its importance, practitioners and researchers alike acknowledge that students have difficulty learning NOS and that this in part reflects how difficult it is to teach. One particularly promising method for teaching (...) NOS involves an explicit and reflective approach using the history of science. The purpose of this study was to determine the influence of a historically based genetics unit on undergraduates’ understanding of NOS. The three-class unit developed for this study introduces students to Mendelian genetics using the story of Gregor Mendel’s work. NOS learning objectives were emphasized through discussion questions and investigations. The unit was administered to undergraduates in an introductory biology course for pre-service elementary teachers. The influence of the unit was determined by students’ responses to the SUSSI instrument, which was administered pre- and post-intervention. In addition, semi-structured interviews were conducted that focused on changes in students’ responses from pre- to post-test. Data collected indicated that students showed improved NOS understanding related to observations, inferences, and the influence of culture on science. (shrink)