This article analyzes the value of geometric models to understand matter with the examples of the Platonic model for the primary four elements (fire, air, water, and earth) and the models of carbon atomic structures in the new science of crystallography. How the geometry of these models is built in order to discover the properties of matter is explained: movement and stability for the primary elements, and hardness, softness and elasticity for the carbon atoms. These geometric models appear to have (...) a double quality: firstly, they exhibit visually the scientific properties of matter, and secondly they give us the possibility to visualize its whole nature. Geometrical models appear to be the expression of the mind in the understanding of physical matter. (shrink)

I investigate the role of stability in cosmology through two episodes from the recent history of cosmology: Einstein’s static universe and Eddington’s demonstration of its instability, and the flatness problem of the hot big bang model and its claimed solution by inflationary theory. These episodes illustrate differing reactions to instability in cosmological models, both positive ones and negative ones. To provide some context to these reactions, I also situate them in relation to perspectives on stability from dynamical systems theory and (...) its epistemology. This reveals, for example, an insistence on stability as an extreme position in relation to the spectrum of physical systems which exhibit degrees of stability and fragility, one which has a pragmatic rationale, but not any deeper one. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

By understanding laws of nature as geometrical rather than linguistic entities, this paper addresses how to describe theory structures and how to evaluate their continuity. Relying on conceptual spaces as a modelling tool, we focus on the conceptual framework an empirical theory presupposes, thus obtain a geometrical representation of a theory’s structure. We stress the relevance of measurement procedures in separating conceptual from empirical structures. This lets our understanding of scientific laws come closer to scientific practice, and avoids a widely (...) recognised deficit in current philosophy of science accounts, namely to risk a collapse of the physical into the mathematical. (shrink)

This article expands on what Bernard Stiegler describes as “The Ordeal of Truth”. Through an evolutionary account of cognition and its exteriorization in human technology, I highlight a recurring tension in philosophy between the “as-if” nature of our models and representations, and the doubt that infects even our most stable understanding of the world. Truth is here associated to the process of metastabilization that characterizes the biological organism. The famous case of Clive Wearing’s severe amnesia, as well as the fictional (...) treatment of amnesia in Christopher Nolan’s Memento, are recalled to highlight the connection between truth and memory, and more importantly the contingent character of the formation of thoughts, and the desperate nature of the construction of truths, through the evolutionarily conditioned “subjective necessity” to link the before and the after, the cause and the effect. I show that this same ordeal is reflected in the philosophy of science in what is known as “pessimistic meta induction”, which I put into dialogue with the growing pessimism about knowledge in contemporary technological culture known as “post truth”. Finally, a possible way out of this pessimism is briefly sketched: a compromise such that, without recourse to objective truth, and in full acceptance of an always already biased relation to the world, future technologies might be designed to incrementally reduce our biases, through an intrinsic mapping of the relative falsity of our models, rather than through some idealized extrinsic relation to an unattainable noumenal real. (shrink)

In this paper I argue that inconsistencies in scientific theories may arise from the type of causality relation they—tacitly or explicitly—embody. All these seemingly different causality relations can be subsumed under a general strategy developed to defeat the paradoxes which inevitably occur in our experience of the real. With respect to this, scientific theories are just a subclass of the larger class of metaphysical theories, construed as theories that attempt to explain a (part of) the world consistently. All metaphysical theories (...) share a common structural backbone specificially designed to defeat paradoxes, their often wildly diverging ontological claims notwithstanding. This common structure shapes the procedures which govern the invention of ideas in the context of such theories, by codifying some onto-logical a priori assumptions regarding the consistency of reality into its bare conceptual framework. Causality plays a key rôle here, because it implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalising power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of ‘system’ and ‘lawful behaviour’. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. The fundamental backbone’s specific gestalt thus functions as a theory’s individual signature and paves the way to a comparative historical approach towards their study. An important part of my paper therefore explores the strong connections between paradoxes as they appear and are dealt with in ancient philosophy and their re-appearance in early modern natural philosophy and science. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results. (shrink)

The development of rigorous foundations of differential calculus in the course of the nineteenth century led to concerns among physicists about its applicability in physics. Through this development, differential calculus was made independent of empirical and intuitive notions of continuity, and based instead on strictly mathematical conditions of continuity. However, for Boltzmann and Poincaré, the applicability of mathematics in physics depended on whether there is a basis in physics, intuition or experience for the fundamental axioms of mathematics—and this meant that (...) to determine the status of differential equations in physics, they had to consider whether there was a justification for these mathematical continuity conditions in physics. For this reason, their ideas about continuity and discreteness in nature were entangled with epistemology and philosophy of mathematics. They reached opposite conclusions: Poincaré argued that physicists must work with a continuous representation of nature, and thus with differential equations, while Boltzmann argued that physicists must ultimately take nature to be discrete. (shrink)

Many philosophers claim to employ intuitions in their philosophical arguments. Others contest that no such intuitions are used frequently or at all in philosophy. This article suggests and defends a conception of intuitions as part of the philosophical method: intuitions are special types of philosophical assumptions to which we are invited to assent, often as premises in argument, that may serve an independent function in philosophical argument and that are not formed through a purely inferential process. A series of philosophical (...) case studies shows that intuitions in these arguments contain the relevant features. The view has implications for philosophical method, offering a compromise between opponents on the divisive debate of the merits of experimental philosophy: experimental philosophy provides an especially useful role in philosophical assumption analysis. (shrink)

Dynamic approaches to understanding probability in the non-fundamental sciences turn on certain properties of physical processes that are apt to produce “probabilistically patterned” outcomes. The dynamic properties on their own, however, seem not quite sufficient to explain the patterns; in addition, some sort of assumption about initial conditions must be made, an assumption that itself typically takes a probabilistic form. How should such a posit be understood? That is the problem of initial conditions. Reichenbach, in his doctoral dissertation, floated a (...) Kantian solution to the problem. In this paper I provide a Reichenbachian alternative. (shrink)

Smartfounding is the opposite of “dumbfounding” introduced by Jonathan Haidt’s research on disgust. Dumbfounders have general competence at thought experiment. However, they are flustered by thought experiments that support repugnant conclusions. Instead of following the supposition wherever it leads, they avoid unsettling implications by adding extraneous information or ignoring stipulated conditions. The dumbfounded commit performance errors, often seeming to regress to the answers of people who lack formal schooling. Smartfounders retain their composure. They practice subversive compliance, obeying the instructions in (...) a way that spoils the aim of the thought experiment. Smartfounders offer a more sophisticated grade of resistance to thought experiment. Whereas the lower grades of resistance are merely fallacious, smartfounding often constitutes a penetrating internal critique of the thought experiment and sometimes of thought experiments in general. (shrink)

Ockham's razor is popularly phrased as a prohibition against multiplying entities beyond necessity. This prohibition should extend to the receptacle for these entities. To state my thesis more positively and precisely, both qualitative and quantitative parsimony apply to space, time, and possibility. All other things equal, we ought to prefer a hypothesis that postulates less space. Smaller is better. Admittedly, scientists are ambivalent about economizing on the void. They praise simplicity. Yet astronomers have a history of helping themselves to as (...) much empty space as they wish—staggering common sense with the vastness of the universe. This largesse is abetted by an aesthetic bias for sublime hypotheses. Whereas abductions based on beauty converge with parsimony, abductions based on the sublime diverge from parsimony. Awe stifles our inner accountant. Both types of abduction are popular. Both are dubious because we lack independent assurance that the universe is beautiful or sublime. The analogy between space and time supports an extension of parsimony to time. Uniformitarian geologists regarded time as a free resource. However, geologists and cosmologists must justify their postulation of an immense past. Possibility follows suit. Contrary to David Lewis, possible worlds should be minimized. Parsimony about possibilities provides a rationale for parsimony about absences. Absences are possibilities that are not actual. So proliferation of absences entails proliferation of possibilities. We should admit only those absences which do explanatory work. (shrink)

I support Rips et al.'s critique of psychology through (1) a complementary argument about the normative, modal, constitutive nature of mathematical principles. I add two reservations about their analysis of mathematical induction, arguing (2) for constructivism against their logicism as to its interpretation and formation in childhood (Smith 2002), and (3) for Piaget's account of reasons in rule learning.

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the (...) minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology. (shrink)

This article addresses the question of the coherence of enactivism as a research perspective by making a case study of enactivism in mathematics education research. Main theoretical directions in mathematics education are reviewed and the history of adoption of concepts from enactivism is described. It is concluded that enactivism offers a ‘grand theory’ that can be brought to bear on most of the phenomena of interest in mathematics education research, and so it provides a sufficient theoretical framework. It has particular (...) strength in describing interactions between cognitive systems, including human beings, human conversations and larger human social systems. Some apparent incoherencies of enactivism in mathematics education and in other fields come from the adoption of parts of enactivism that are then grafted onto incompatible theories. However, another significant source of incoherence is the inadequacy of Maturana’s definition of a social system and the lack of a generally agreed upon alternative. (shrink)

Despite the conclusions from the contemporary philosophy of science, many economists cherish the ideal of positive science. Therefore, value-free economics is still the central paradigm in economics. The first aim of the paper is to investigate economics' axiomatic assumptions from an epistemological perspective. The critical analysis of the literature shows that the positive-normative dichotomy is exaggerated. Moreover, value-free economics is based on normative foundations that have a negative impact on individuals and society. The paper's second aim is to show that (...) economics' normativity is not a problem because the discussion concerning values is possible and unavoidable. In this context, Weber and other methodologists are investigated. The conclusion of the paper is that science can thrive without strict methodological rules thanks to institutional mechanisms. Therefore, economists could learn from artists who accept the world without absolute rules. This perspective opens the possibility for methodological pluralism and normative approaches. (shrink)

The aim of psychology is first of all to describe how we think, or the flow of our consciousness, and then to sum up the facts in terms of principles, generalizations, or “laws” which “govern” our thinking. These laws must enable us to foretell what a man will think and how he will act when we know his environment and the state of his thought at any given moment, provided that no unforeseeable exercise of free will intervenes. Psychology has also (...) to inquire whether interventions of “free will” ever do occur and, if they do, what is their nature and scope. (shrink)

Cet essai cherche à faire de la complémentarité entre énergie-idée, structure et fonction, et autres couples de concepts, la base d'une nouvelle ontologie qui puisse résoudre les conflits entre les pôles de description « mental » et « physique », entre la vérité mathématique et la vérité empirique et entre la mécanique quantique et la théorie de la relativité comme formes rivales d'explication scientifique. L'auteur y plaide en faveur de la fermeture déductive de l'univers à la lumière de la relation (...) logique entre le nombre et l'alphabet . Il y présente la réalité et l'autonomie conceptuelle des idées comme une alternative au platonisme et à d'autres idéologies. Il dispose de fausses interprétations, malentendus, et objections au moyen du dialogue.Conjugacy between energy-idea, structure and function, and other concept pairs is made the basis for a new ontology which resolves the conflicts between "mental" and "physical" poles of description, between mathematical and empirical truth, and between quantum mechanics and relativity theory as rival forms of scientific explanation. The deductive closure of the universe is argued for, in light of the logical connection between number and the alphabet . The reality and the conceptual autonomy of ideas are presented as an alternative to Platonism and other ideologies. Mistaken interpretations, misunderstandings and objections are handled through the medium of dialogue. (shrink)

This paper addresses the definition and the operational use of intuitions in philosophical methods in the form of a research study encompassing several regions of the globe, involving 282 philosophers from a wide array of academic backgrounds and areas of specialisation. The authors tested whether philosophers agree on the conceptual definition and the operational use of intuitions, and investigated whether specific demographic variables and philosophical specialisation influence how philosophers define and use intuitions. The results obtained point to a number of (...) significant findings, including that philosophers distinguish between intuitions used to formulate (discovery) and to test (justification) philosophical theory. The survey results suggest that strategies implemented to characterise philosophical intuition are not well motivated since, even though philosophers do not agree on a single account of intuition, they fail to capture a preferred usage of intuitions as aspects of discovery. The quantitative summary of survey findings informs the debate on this topic, and advances more defined routes for subsequent approaches to the study of intuitions. (shrink)

Hans Reichenbach's 1928 thesis of the relativity of geometry has been misunderstood as the statement that the geometrical structure of space can be described in different languages. In this interpretation the thesis becomes an instance of trivial semantical conventionalism, as Grünbaum calls it. To understand Reichenbach correctly, we have to interpret it in the light of the linguistic turn, the transition from thought oriented philosophy to language oriented philosophy, which mainly took place in the first decades of our century. Reichenbach (...) — as Poincaré before him — is undermining the prejudice of thought oriented philosophy, that two propositions have different factual content, if they are associated with different ideas in our mind. Thus Reichenbach prepared the change to language oriented philosophy, which he also accepted later. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...) This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU. (shrink)

In the discourse of the knowledge-based economy, the link between creativity and innovation is usually taken for granted. However, not only is this link of recent date, it joins together two diametrically opposed concepts: the economic concept of innovation and the humanistic concept of creativity. In research too, there is a lack of enquiry into the nature of the processes of creativity and innovation and into how these processes are similar yet different. Building on the original insights of Henri Poincaré (...) and Joseph Schumpeter, I present the hypothesis that creativity and innovation are similar in nature, not in regard to knowledge or skills, but in that both are ways of relating to the novel. My findings suggest, meanwhile, that creativity and innovation differ in that creativity is about experiencing the novel in the creative process, whereas innovation is about willing the novel as a strategy for the innovative process. Furthermore, my findings suggest that there exists a structural similarity between creativity and innovation and Aristotle’s concepts of practical and philosophical wisdom, in that innovation is a form of practical wisdom in regard to the novel, whereas creativity is a form of philosophical wisdom in regard to the novel. (shrink)

The aim of this article is to contribute to responsible innovation by developing a conceptual framework for the processes of creativity and innovation. The hypothesis is that creative and innovative processes are similar in that both are affective in nature. I develop this conceptual framework through an interpretation of the insights of Henri Poincaré’s notion of the ‘four stages’ in the creative process and Joseph Schumpeter’s notion of the entrepreneur. Building on this framework, I analyze the creative and innovative practices (...) of the film director Lars von Trier and the entrepreneur Steve Jobs. The interpretation and analysis suggest that the processes of creativity and innovation are similar in nature in that both are based on the moods of disturbance and enthusiasm; but that they differ in that creativity is based on the feelings of interest and irritation, whereas innovation is based on the feelings of desire and anger. In the conclusion I discuss the implications of this for responsible innovation with regard to the social aspect of resistance towards innovation and the ethical aspects of anger in entrepreneurial leadership. (shrink)

A mahāvidyā inference is used for establishing another inference. Its Reason is normally an omnipresent property. Its Target is defined in terms of a general feature that is satisfied by different properties in different cases. It assumes that there is no case that has the absence of its Target. The main defect of a mahāvidyā inference μ is a counterbalancing inference that can be formed by a little modification of μ. The discovery of its counterbalancing inference can invalidate such an (...) inference. This paper will argue that Cantor’s diagonal argument too shares some features of the mahāvidyā inference. A diagonal argument has a counterbalanced statement. Its main defect is its counterbalancing inference. Apart from presenting an epistemological perspective that explains the disquiet over Cantor’s proof, this paper would show that both the mahāvidyā and diagonal argument formally contain their own invalidators. (shrink)

This paper offers a novel way of reconstructing conceptual change in empirical theories. Changes occur in terms of the structure of the dimensions—that is to say, the conceptual spaces—underlying the conceptual framework within which a given theory is formulated. Five types of changes are identified: (1) addition or deletion of special laws, (2) change in scale or metric, (3) change in the importance of dimensions, (4) change in the separability of dimensions, and (5) addition or deletion of dimensions. Given this (...) classification, the conceptual development of empirical theories becomes more gradual and rationalizable. Only the most extreme type—replacement of dimensions—comes close to a revolution. The five types are exemplified and applied in a case study on the development within physics from the original Newtonian mechanics to special relativity theory. (shrink)

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.

The paper reviews criteria which have been used to distinguish science from nonscience and from pseudo–science, and it examines the extent to which they can usefully be applied to “creation science.” These criteria do not force a clear decision, especially as creation science resembles important eighteenth–century forms of orthodox science. Nevertheless, the proponents of creation science may be accused of pious fraud in failing to concede in their political battles that their “science” is tentative and tendentious and will continue to (...) be so while it remains archaic and poorly integrated into the rest of science. (shrink)

This article examines how experimental psychology experienced a revolution as cognitive science replaced behaviorism in the mid-20th century. This transition in the scientific account of human nature involved making normal what had once been normative: borrowing ideas of democratic thinking from political culture and conceptions of good thinking from philosophy of science to describe humans as active, creatively thinking beings, rather than as organisms that simply respond to environmental conditions. Reflexive social and intellectual practices were central to this process as (...) cognitive scientists used anti-positivist philosophy of science simultaneously to justify their own work as valid and also as a model of human thinking. In the process, the normative philosophy of science or ‘good academic thinking’ that cognitive scientists used to reshape the discipline of psychology and characterize themselves became, at the same time, the descriptive model of human nature. (shrink)

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which hypothesis to (...) consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research. (shrink)

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

This article reviews recent findings indicating some common brain processes during dissociative states and dreaming with the aim to outline a perspective that neural chaotic states during dreaming can be closely related to dissociative states that may manifest in dreams scenery. These data are in agreement with various clinical findings that dissociated states can be projected into the “dream scenery” in REM sleep periods and dreams may represent their specific interactions that may uncover unusual psychological potential of creativity in psychotherapy, (...) art and scientific discoveries. (shrink)

It would seem that an epistemic framework can be justified only by means of a non-circular argument that establishes its truth-conduciveness. The problem of epistemic circularity suggests that no such argument is possible. Externalists and particularists have addressed the problem of scepticism by claiming that epistemically circular arguments can establish the truth-conduciveness of a framework’s epistemic methods. However, since these arguments are available for a good many frameworks, this response does nothing to answer the threat of epistemic relativism. The purpose (...) of this paper is to offer a non-circular argument in favour of naturalistic epistemic frameworks, and therefore against epistemic relativism. The argument establishes that non-naturalistic frameworks depend on naturalistic methods for their application, and therefore, they cannot be more truth-conducive than naturalistic frameworks. This argument does not establish that naturalistic frameworks are truth-conducive, but it does provide a ra.. (shrink)

In this paper, I try to refute the Kantian a priorism. At the same time, I try to explain the existence of an a priori concerning space and time on the basis of contemporary neuro-physiology. This a priori is the opposite of the a-historical a priori of Kant. Concerning space and time, I argue that relativity concords with the philosophical thesis that space and time are forms of existence of matter. On the basis of this ontological principle, I support that (...) by accepting the existence of local absolute systems of reference, it is possible to explain some paradoxes of special relativity and at the same time to refute the relativism related to the theory of Einstein. (shrink)

When philosophers discuss the possibility of machines making scientific discoveries, they typically focus on discoveries in physics, biology, chemistry and mathematics. Observing the rapid increase of computer-use in science, however, it becomes natural to ask whether there are any scientific domains out of reach for machine discovery. For example, could machines also make discoveries in qualitative social science? Is there something about humans that makes us uniquely suited to studying humans? Is there something about machines that would bar them from (...) such activity? A close look at the methodology of interpretive social science reveals several abilities necessary to make a social scientific discovery, and one capacity necessary to possess any of them is imagination. For machines to make discoveries in social science, therefore, they must possess imagination algorithms. (shrink)

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

Arguably no concept is more fundamental to science than that of causality, for investigations into cases of existence, persistence, and change in the natural world are largely investigations into the causes of these phenomena. Yet the metaphysics and epistemology of causality remain unclear. For example, the ontological categories of the causal relata have been taken to be objects (Hume 1739), events (Davidson 1967), properties (Armstrong 1978), processes (Salmon 1984), variables (Hitchcock 1993), and facts (Mellor 1995). (For convenience, causes and effects (...) will usually be understood as events in what follows.) Complicating matters, causal relations may be singular (Socrates’s drinking hemlock caused Socrates’s death) or general (Drinking hemlock causes death); hence the relata might be tokens (e.g., instances of properties) or types (e.g., types of events) of the category in question. Other questions up for grabs are: Are singular causes metaphysically and/or epistemologically prior to general causes or vice versa (or neither)? What grounds the intuitive asymmetry of the causal relation? Are macro-causal relations reducible to micro-causal relations? And perhaps most importantly: Are causal facts (e.g., the holding of causal relations) reducible to non-causal facts (e.g., the holding of certain spatiotemporal relations)? (shrink)

Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogeneous, simple units, the atoms or cells. At each time unit, the cells instantiate one of a finite set of states. They evolve in parallel at discrete time steps, following (...) state update functions or dynamical transition rules: the update of a cell state obtains by taking into account the states of cells in its local neighborhood (there are, therefore, no actions at a distance). Secondly, CA are abstract, as they can be specified in purely mathematical terms and implemented in physical structures. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems. Despite functioning in a different way from traditional, Turing machine-like devices, CA with suitable rules can emulate a universal Turing machine, and therefore compute, given Turing's Thesis, anything computable.... (shrink)

REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.

Many philosophers regard the persistence of philosophical disputes as symptomatic of overly ambitious, ill-founded intellectual projects. There are indeed strong reasons to believe that persistent disputes in philosophy (and more generally in the discourse at large) are pointless. We call this the pessimistic view of the nature of philosophical disputes. In order to respond to the pessimistic view, we articulate the supporting reasons and provide a precise formulation in terms of the idea that the best explanation of persistent disputes entails (...) that they are pointless. We then show how to answer the pessimistic argument. Taking a well-known mathematical controversy as our paradigm example, we argue that some persistent disputes reflect substantive disagreements at the “meta-analytic” level, i.e., disagreements about the best way, among quite different candidates, to understand the topic at issue, and the best associated cluster of analytic truths one should accept concerning it. Moreover, our concrete example shows that such meta-analytic disagreements can in principle be settled and yield a genuine theoretical (as opposed to merely pragmatic) breakthrough. We conclude optimistically that persistent disputes can be an important means of fostering epistemic progress. (shrink)

The subject of this essay-based dissertation is Hume’s natural philosophy. The dissertation consists of four separate essays and an introduction. These essays do not only treat Hume’s views on the topic of natural philosophy, but his views are placed into a broader context of history of philosophy and science, physics in particular. The introductory section outlines the historical context, shows how the individual essays are connected, expounds what kind of research methodology has been used, and encapsulates the research contributions of (...) the essays. The first essay treats Newton’s experimentalist methodology in gravity research and its relation to Hume’s causal philosophy. It is argued that Hume does not see the relation of cause and effect as being founded on a priori reasoning, similar to the way in which Newton criticized non-empirical hypotheses about the causal properties of gravity. Contrary to Hume’s rules of causation, the universal law does not include a reference either to contiguity or succession, but Hume accepts it in interpreting the force and the law of gravity instrumentally. The second article considers Newtonian and non-Newtonian elements in Hume more broadly. He is sympathetic to many prominently Newtonian themes in natural philosophy, such as experimentalism, critique of hypotheses, inductive proof, and the critique of Leibnizian principles of sufficient reason and intelligibility. However, Hume is not a Newtonian philosopher in many respects: his conceptions regarding space and time, the vacuum, the specifics of causation, the status of mechanism, and the reality of forces differ markedly from Newton’s related conceptions. The third article focuses on Hume’s Fork and the proper epistemic status of propositions of mixed mathematics. It is shown that the epistemic status of propositions of mixed mathematics, such as those concerning laws of nature, is that of matters of fact. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle. The fourth article analyzes Einstein’s acknowledgement of Hume regarding special relativity. The views of the scientist and the philosopher are juxtaposed, and it is argued that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts and a relationist ontology regarding space and time. (shrink)

I examined the proposition that there are psychological limits on what scientific problems can be solved, and that these limits may be based on a failure to be able to produce imagable, observation-based models for any possible solution, a position suggested by philosopher Colin McGinn in an argument attempting to prove that the mind-body problem is unsolvable. I examined another likely candidate for an unsolvable problem -- the ultimate origin of the universe (i.e., what might have preceded the Big Bang (...) or any other starting point; why there should be something rather than nothing) -- by exploring the reasoning of physicists about this problem and measuring visual imagery frequency and vividness, with the expectation that those who most believed the problem unsolvable would be more frequent/vivid imagers and therefore more affected by the apparent impossibility of producing an imagable solution. Eight physicists were interviewed and imagery frequency and vividness measurements performed using Cohen & Saslona's IDQ-IHS and Marks's VVIQ, respectively. All subjects considered the problem unsolvable within today's physics and all but one thought the problem still meaningful, though none were optimistic about a solution. The one subject who dismissed the problem had the lowest imagery frequency score, and there was also a significant rank order correlation (r = 0.83, p < .02) between degree of belief in problem unsolvability (extended to include viewing the problem as meaningful and not already solved) and a composite imagery frequency/vividness score, though the sample was too small to control for some possible confounding factors. (shrink)

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...) that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed. (shrink)

From the perspective of nonlinear science, it is argued that one may accept physicalism and reject substance dualism without being forced into reductionism. This permits a property dualism under which biological and mental phenomena may emerge from intricate positive feedback networks, involving many levels of both the biological and cognitive hierarchies.