I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

Some philosophers have laid down rather severe strictures on whether there can be thought without language. Wittgenstein asserted that ‘the limits of language…mean the limits of my world’. Davidson has argued that ‘a creature cannot have thoughts unless it is an interpreter of the speech of another’. Dummett has interpreted some pronouncements as meaning that ‘the study of thought is to be sharply distinguished from the study of the psychological processes of thinking and…the only proper method of analysing thought consists (...) in the analysis of language’. And there is also the position that thought has its own language that might exist even prior to or in the absence of natural language. But here I am going to concentrate on what might be possible in the absence of natural language. I do not know what it would mean to consider thinking in the absence of its own intrinsic language, a language of thought, if the two always co-exist. (shrink)

A description of the author's application of protocol analysis in the teaching of philosophy.

The concept of creativity is a central one in folk psychological explanation and has long been prominent in philosophical debates about the nature of art, genius, and the imagination. The scientific investigation of creativity in humans is also well established, and there has been increasing interest in the question of whether the concept can be rigorously applied to non-human animals. In this paper, I argue that such applications face serious challenges of both a conceptual and methodological character, reflecting deep controversies (...) within both philosophy and psychology concerning how to define and apply the concept of creativity. After providing a brief review of some of the leading theories of creativity and discussing some of the strongest putative cases of creative intelligence in non-human animals, I examine some of the more worrisome difficulties faced by attempts to use these theories to answer the question of whether animals are truly creative. I conclude by examining how we might overcome them, and suggest that one approach worth taking seriously is to adopt what I term a Strong Rejectionist view of creativity, eschewing use of the term entirely in the scientific study of comparative cognition. (shrink)

_ Source: _Volume 11, Issue 2, pp 117 - 148 The paper examines the problem of self-knowledge from the perspectives of Plotinus and the Indian Advaita school. Analyzing the subject-object relation, I show that according to both Plotinus and Advaita thinkers, full self-knowledge demands complete absence of otherness. Plotinus argues that if self-consciousness is divided into subject-object relation then one will know oneself as contemplated but not as contemplating and no real self-knowledge obtains in this case. Śaṅkara, who constitutes an (...) important representative of Advaita thought, points out that the self cannot know itself as an object because what is called an object to be known becomes established when it is separated from the self, the subject. I argue that at the level of the One, similar to the state of _ātman_ consciousness in Advaita framework, the soul experiences itself in expansive non-dual consciousness. Lastly, I examine the role of non-duality as the foundation of knowledge. (shrink)

Many mathematicians have a rich internal world of mental imagery. Using elementary mathematical skills, this study probes the mathematical imagination's sensorimotor foundations. Mental imagery is perturbed using body position: having the head and vestibular system in different positions with respect to gravity. No two mathematicians described the same imagery. Eight out of 11 habitually visualize, one uses sensorimotor imagery, and two do not habitually used mental imagery. Imagery was both intentional and partly autonomous. For example, coordinate planes rotated, drifted, wobbled, (...) or slid down from vertical to horizontal. Parabolae slid into place or, on one side, a parabola arm reached upward in gravity. The sensorimotor foundation of imagery was evidenced in several ways. The imagery was placed with respect to the body. Further, the imagery had a variety of relationships to the body, such as the body being the coordinate system or the coordinate system being placed in front of the eyes for easy viewing by the mind's eye. The mind's eye, mind's arm, and awareness almost always obeyed the geometry of the real eye and arm. The imagery and body behaved as a dyad, so that the imagery moved or placed itself for the convenience of the mind's eye or arm, which in turn moved to follow the imagery. With eyes closed, participants created a peripersonal imagery space, along with the peripersonal space of the unseen environment. Although mathematics is fundamentally abstract, imagery was sometimes concrete or used a concrete substrate or was placed to avoid being inside concrete objects, such as furniture. Mathematicians varied in the numbers of components of mental imagery and the ways they interacted. The autonomy of the imagery was sometimes of mathematical interest, suggesting that the interaction of imagery habits and autonomy can be a source of mathematical creativity. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

Perhaps a part of what makes expertise so inspiring to the curious researcher is the possibility of appropriating the structural components of skilled action to draw a roadmap towards their achievement that anyone might be able to follow. Accordingly, the purpose of this essay is to shed light upon the role that creativity plays in the production and environment of skilled action to that foregoing end. In doing so, I suggest that the lessons to be learned from recent empirical research (...) on creativity has much to offer to the cognitive science of skill and expertise. Experts are able to bring their intelligence to bear in controlling fast and seemingly automatic actions by utilizing a form of control often called ‘intelligent automaticity.’ In this spirit, I argue that the environment of intelligently automatic action control curates a similarly ideal environment for the processes of creativity. Moreover, insofar as creativity is ideally operative within the environment of expert action control, I argue further that creativity functions as one representative form of ‘intelligence’ embedded within otherwise fluid, and automatic expert actions. Creativity is able to do so even without conscious representation through the powers of incubated cognition. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. (...) The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures. (shrink)

Journal of Philosophy of Education, EarlyView.

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)

The aim of the article is to contribute to the conceptualization of creativity in education. The article makes use of the self-Bildung perspective, which is an up-to-date version of the Neo-humanistic notion of the formation of the personality in order to interpret the original notion of the ‘four stages’ of the creative process. The conclusion is that in this perspective creativity can be understood as a state of transcendence and a practice of taste. Finally, the article seeks to sketch out (...) a suggestion of the creative capacities that are important to cultivate in education. (shrink)

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I (...) will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)

The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman and Olbrechts-Tyteca which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. Given this framework, we can better understand (...) the introduction of proof methods based on the mathematician’s likely universal audience. I examine a case study from Alexander and Briggs’s work on knot invariants to show that we can fruitfully reconstruct mathematical methods in terms of audiences. (shrink)

In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within (...) the closed world of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any necessary dynamic of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto. (shrink)

A sourcebook/textbook on the problem of free will and determinism. Contains a history of the free will problem, a taxonomy of current free will positions, the standard argument against free will, the physics, biology, and neuroscience of free will, the most plausible and practical solution of the problem, and reviews of the work of the leading determinist Ted Honderich, the leading libertarian Robert Kane, the well-known compatibilist Daniel Dennett, and the determinism-agnostic Alfred Mele.

Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...) assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​. (shrink)

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61students in six small sections of a “bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate the top-level logical structure of a proof. For simplified informal (...) calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or evaluate their validity. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

The main purpose of this dissertation is to examine critically and discuss the role of imagination in science and religion, with particular emphasis on its possible epistemic, creative, and meaning-making functions. In order to answer my research questions, I apply theories and concepts from contemporary philosophy of mind on scientific and religious practices. This framework allows me to explore the mental state of imagination, not as an isolated phenomenon but, rather, as one of many mental states that co-exist and interplay (...) in our cogntive architecture. Based on the philosophical discourse of philosophy of mind, four types of imagination are indentified and conceptualized: sensory, propositional, experiential, and creative imagination. These categories are then employed on five phenomena that can be found in scientific and religious environments: metaphors, models, thought experiments, aspect perception, and - in the religious case - rituals. In relation to the concept of religious "seeing" I consider how imaginings may influence visionary experiences and visualization, and compare these phenomena with cases of scientific visualization and eureka experiences. In regard to scientific and religious models, a distinction is made between, on the one hand, two notions of truth and, on the other hand, truth-independent meaning-making. In light of these categories, I differentiate between, and critically discuss, the use of imagination in doxastic, non-doxastic and fictionalist accounts. In light of this investigation, I formulate and defend the position of interactivism, which acknowledges a constant interplay between different attitudes and mental states. In my examination of rituals and scientific and religious thought experiments, special attention is given to the mental capacity to recreate the experiences that are entailed in an imagined scenario. At the end of the investigation, I consider the possible impact that my study might have on how we view science and religion as well as the dialogue bewteen these two fields. (shrink)

The main thrust of the argument of this thesis is to show the possibility of articulating a method of construction or of synthesis--as against the most common method of analysis or division--which has always been (so we shall argue) a necessary component of scientific theorization. This method will be shown to be based on a fundamental synthetic logical relation of thought, that we shall call inversion--to be understood as a species of logical opposition, and as one of the basic monadic (...) logical operators. Thus the major objective of this thesis is to This thesis can be viewed as a response to Larry Laudan's challenge, which is based on the claim that ``the case has yet to be made that the rules governing the techniques whereby theories are invented (if any such rules there be) are the sorts of things that philosophers should claim any interest in or competence at.'' The challenge itself would be to show that the logic of discovery (if at all formulatable) performs the epistemological role of the justification of scientific theories. We propose to meet this challenge head on: a) by suggesting precisely how such a logic would be formulated; b) by demonstrating its epistemological relevance (in the context of justification) and c) by showing that a) and b) can be carried out without sacrificing the fallibilist view of scientific knowledge. OBJECTIVES: We have set three successive objectives: one general, one specific, and one sub-specific, each one related to the other in that very order. (A) The general objective is to indicate the clear possibility of renovating the traditional analytico-synthetic epistemology. By realizing this objective, we attempt to widen the scope of scientific reason or rationality, which for some time now has perniciously been dominated by pure analytic reason alone. In order to achieve this end we need to show specifically that there exists the possibility of articulating a synthetic (constructive) logic/reason, which has been considered by most mainstream thinkers either as not articulatable, or simply non-existent. (B) The second (specific) task is to respond to the challenge of Larry Laudan by demonstrating the possibility of an epistemologically significant generativism. In this context we will argue that this generativism, which is our suggested alternative, and the simplified structuralist and semantic view of scientific theories, mutually reinforce each other to form a single coherent foundation for the renovated analytico-synthetic methodological framework. (C) The third (sub-specific) objective, accordingly, is to show the possibility of articulating a synthetic logic that could guide us in understanding the process of theorization. This is realized by proposing the foundations for developing a logic of inversion, which represents the pattern of synthetic reason in the process of constructing scientific definitions. (shrink)

Many traditional theories of creativity put heavy emphasis on an incubation stage in creative cognitive processes. The basic phenomenon is a familiar one: we are working on a task or problem, we leave it aside for some period of time, and when we return attention to the task we have some new insight that services completion of the task. This feature, combined with other ostensibly mysterious features of creativity, has discouraged naturalists from theorizing creativity. This avoidance is misguided: we can (...) maintain unconscious incubated cognition as (sometimes) part of the creative process and we can explain it in scientifically responsible ways. This paper, focusing on the effects of attention on the functional networking of the brain, attempts just such an explanation. It also serves to assuage the naturalist's scepticism about other features of creative cognition. The broad upshot, one would hope, is that philosophers of mind and cognitive scientists return some attention to the long neglected topic of creativity. (shrink)

In earlier papers I described the conscious electromagnetic information (CEMI) field theory, which claimed that the substrate of consciousness is the brain’s electromagnetic (EM) field. I here further explore this theory by examining the properties and dynamics of the information underlying meaning in consciousness. I argue that meaning suffers from a binding problem, analogous to the binding problem described for visual perception, and describe how the gestalt (holistic) properties of meaning give rise to this binding problem. To clarify the role (...) of information in conscious meaning, I differentiate between extrinsic information that is symbolic and arbitrary, and intrinsic information, which preserves structural aspects of the represented object and thereby maintains some gestalt properties of the represented object. I contrast the requirement for a decoding process to extract meaning from extrinsic information, whereas meaning is intrinsic to the structure of the gestalt intrinsic information and does not require decoding. I thereby argue that to avoid the necessity of a decoding homunculus, conscious meaning must be encoded intrinsically — as gestalt information — in the brain. Moreover, I identify fields as the only plausible substrate for encoding gestalt intrinsic information and argue that the binding problem of meaning can only be solved by grounding meaning in this field-based gestalt information. I examine possible substrates for gestalt information in the brain and conclude that the only plausible substrate is the CEMI field. (shrink)

Research into two-stage models of “free will” – first “free” random generation of alternative possibilities, followed by “willed” adequately determined decisions consistent with character, values, and desires – suggests that William James was in 1884 the first of a dozen philosophers and scientists to propose such a two-stage model for free will. We review the later work to establish James’s priority. By limiting chance to the generation of alternative possibilities, James was the first to overcome the standard two-part argument against (...) free will, i.e., that the will is either determined or random. James gave it elements of both, to establish freedom but preserve responsibility. We show that James was influenced by Darwin’s model of natural selection, as were most recent thinkers with a two-stage model. In view of James’s famous decision to make his first act of freedom a choice to believe that his will is free, it is most fitting to celebrate James’s priority in the free will debates by naming the two-stage model – first chance, then choice -“Jamesian” free will. (shrink)