The book delves into mathematical mindsets, a current approach to understanding mathematical identities, as well as success and failure in mathematics.

A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is (...) shown that Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)

Constructivist and embodied theories of learning each focus on action as the basis for cognition. However, in restricting action to sensorimotor activity, some embodied perspectives eschew ….

This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida (...) and de Man. (shrink)

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue (...) that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it. (shrink)

This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of (...) cognitive analyses of historical developments of mathematics has been primarily on the former, even if they claim to be about the latter. (shrink)

Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.

Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for (...) example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications (For instance, Mycielski’s approach). Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration. (shrink)

Metaphors and models involve correspondences between events in separate domains. They differ in the form and precision of how the correspondences are expressed. Examples include correspondences between phylogenic and ontogenic selection, and wave and particle metaphors of the mathematics of quantum physics. An implication is that the target article's metaphors of resistance to change may have heuristic advantages over those of momentum.

How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of (...) mathematics. By taking these metaphors seriously we attain a new vantage point for understanding historical changes in conceptions of mathematics. (shrink)

Context: How can an enactive approach to the teaching and learning of mathematics be implemented, which fosters mathematical thinking, making intensive use of metaphorizing and taking into account ….

RésuméDans leur livre récent, George Lakoff et Rafael Núñez se livrent à une critique naturaliste soutenue du platonisme traditionnel concernant les entités mathématiques. Ils affirment que des résultats récents en sciences cognitives démontrent qu'il est faux. En particulier, ils estiment que la découverte que la cognition mathématique s'appuie pour une large part sur les métaphores conceptuelles est incompatible avec le platonisme. Nous montrons ici que tel n'est pas le cas. Nous examinons et rejetons également quelques arguments philosophiques que formulent Lakoff (...) et Núñez contre le platonisme. Enfin, nous lions leur hostilité au platonisme à certaines opinions sur les ramifications politiques de celui-ci. Pour conclure, nous faisons valoir que ces opinions sont sans fondements et présentent une image distordue des dimensions politiques du platonisme. (shrink)

The ArgumentWhen economists report on research using mathematical models, they use a literary form similar to the experimental report in the laboratory sciences. This form consists of a narrative of a series of events, with a clear temporal segregation of the agency of the author and the agency of the objects of study. Existing explanations of this literary form treat it as a rhetorical device that either conceals the agency of the author in constructing and interpreting the findings, or (...) simply appropriates the appearance of accepted scientific method. This article — based on analysis of a research program in economics, a single article that issued from that program, and in-depth interviews with the authors — proposes an alternate interpretation. Drawing on the praxeological “laboratory studies” tradition in science studies, we treat work with mathematical models as involving the interaction of economists with objects that act independently of the analyst's will. The clear separation of the economist's and the models agency, as depicted in the published report, is not the result of a rhetorical rewriting of actual events, but is a practical accomplishment. Every step in the analytical work that preceded the paper is devoted to developing a procedure in which the economists' agency will be completely accountable in terms of accepted practices, and the performance of the model will be distinct and compelling. (shrink)

Contemporary analytic philosophy recognizes few principled constraints on its subject matter. When other disciplines also lay claim to a particular topic, however, important questions arise concerning the relation between these other disciplines and philosophy. A case in point is mathematics: traditional philosophy of mathematics defines a set of problems and certain general answers to those problems. However, mathematics is a subject matter that can be studied in many other ways: historically, sociologically, or even aesthetically, for example. Given this, we may (...) ask what implications discoveries in these disciplines have for the philosophy of mathematics. (shrink)

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. & Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, M. (2003a). Semiotica, 145, (...) 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations. (shrink)

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism (...) on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of ‘grasping’ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes ‘grasping’ more as theoretical activity than as a kind of inner mental ‘seeing’. (shrink)

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is (...) on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts. (shrink)

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least, scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track (...) with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts. (shrink)

The enigmatic thought of Charles S. Peirce (1839-1914), considered by many to be one of the great philosophers of all time, involves inquiry not only into virtually all branches and sources of modern semiotics, physics, cognitive sciences, and mathematics, but also logic, which he understood to be the only useful approach to the riddle of reality. This book represents an attempt to outline an analytical method based on Charles Peirce's least explored branch of philosophy, which is his evolutionary cosmology, and (...) his notion that the universe as made of an 'effete mind.' The chief argument conceives of human discourse as a giant metaphor in regard to outside reality. The metaphors arise in our imagination as lightning-fast schemes of acting, speaking, or thinking. To prove this, each chapter will present a well-known metaphor and explain how it is unfolded and conceptualized according to the new method for revealing meaning. This original work will interest students and scholars in many fields including semiotics, linguistics and philosophy. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporary (...) geometric morphometrics, the notion of a morphospace is based on the Euclidean tangent space to Kendall’s shape space, which is a Riemannian manifold. Many more classical morphospaces, such as Raup’s space of coiled shells, lack these metric properties, e.g., due to incommensurably scaled variables, so that these morphospaces typically are affine vector spaces. Other notions of a morphospace, like Thomas and Reif’s skeleton space, may not give rise to a quantitative measure of similarity at all. Such spaces can often be characterized in terms of topological or pretopological spaces. (shrink)

Rips et al. appear to discuss, and then dismiss with counterexamples, the brain-based theory of mathematical cognition given in Lakoff and Nez (2000). Instead, they present another theory of their own that they correctly dismiss. Our theory is based on neural learning. Rips et al. misrepresent our theory as being directly about real-world experience and mappings directly from that experience.

The enigmatic thought of Charles S. Peirce, considered by many to be one of the great philosophers of all time, involves inquiry not only into virtually all branches and sources of modern semiotics, physics, cognitive sciences, and mathematics, but also logic, which he understood to be the only useful approach to the riddle of reality. This book represents an attempt to outline an analytical method based on Charles Peirce’s least explored branch of philosophy, which is his evolutionary cosmology, and his (...) notion that the universe is made of an ‘effete mind.’ The chief argument conceives of human discourse as a giant metaphor in regard to outside reality. The metaphors arise in our imagination as lightning-fast schemes for acting, speaking, or thinking. To illustrate this, each chapter will present a well-known metaphor and explain how it is unfolded and conceptualized according to the new method for revealing meaning. This original work will interest students and scholars in many fields including semiotics, linguistics and philosophy. (shrink)

This chapter contains sections titled: Models Analogy Metaphor Metaphorical Models Current Issues.

This article shows how Arab students at an Arab college in Israel, majoring in teaching of mathematics, English, and science, rely on metaphor as an important rhetorical tool for the advancement of their ideological positions and for criticism of the policies of the Israeli government, which discriminates against and disenfranchises Arab Israelis. The underlying hypothesis of the article is that the way Arab students in Israel use metaphor in their writing has unique rhetorical aspects that help to sharpen their message (...) in pursuit of the broader goal of emphasizing Arab Israelis’ suffering and changing for the better the Israeli government’s discriminatory patterns of action against them. (shrink)

Some ideas seem to possess a disproportionate ability to lead to new insights, new discoveries, new ideas, and even entirely new ways of thinking. Such ideas are said to be fruitful. Looking across the history of science and mathematics, we see creative minds preoccupied with the search for ideas of this kind. More precious than truth, fruitful ideas provide those in pursuit of knowledge with a seemingly bottomless well of innovation from which to draw as they attempt to solve new (...) problems and to refine solutions to old ones. Professional researchers have a nose for these ideas. They often can see in an instant that some way of approaching a problem will eventually result in a solution to it and to a whole host of other problems, all of which suddenly seem related. In this book, Chris Haufe explains how these ideas are detected and developed into large-scale frameworks for research. He argues for a philosophical perspective on scientific knowledge that places the search for fruitfulness at the heart of the scientific enterprise. (shrink)

Excerpts from Chapters 1 and 3 of New Maps for Old: Explorations in Science and Religion (Gerhart and Russell 2001) explore the ramifications of metaphoric process for changes in thinking, especially those changes that lead to a new understanding of our world. Examples are provided from science, from religion, and from science and religion together. In excerpts from Chapter 8, a double analogy—theology is to science as science is to mathematics—is proposed for better understanding the contemporary relationship between science and (...) religion. A conservation of epistemological sufficiency is disclosed as one moves from mathematics to empirical science to theology—a move from one discipline to another that involves a sacrifice of one aspect of thought to gain another. (shrink)

ABSTRACTWhile metaphor researchers commonly use the word “mapping” in explanations of various types of figurative language, there is a lack of recognition that the term is itself metaphorical. In fact, the term has two metaphor-based working definitions, the more commonly cited being that relating to mathematical set theory and the less common definition originating in cognitive neuroscience. Perhaps not coincidentally, terminological inconsistencies relating to mapping have led to theoretical problems both for single-domain theories of metonymy and attempts to examine (...) Lakoff’s invariance hypothesis. This article will assert that expressing metonymic connectivity using the neuroscience term “binding” will both eliminate unnecessary theoretical confusion for cognitive linguists and provide common terminology to facilitate productive communication between cognitive linguists and cognitive neuroscientists. (shrink)

In summary, many extant definitions of the niche concept are based on the geometric metaphor which represents the niche as an object embedded in a geometric space. There are several difficulties with this approach; the activities of organisms are not fully described, certain attributes of the functional aspect of the niche are not represented, the life cycles of organisms are not described, and the heuristic value of the concept diminishes with increasing dimensionality.An alternative and complementary approach to the niche is (...) the linguistic metaphor. This approach is based on the explanatory objective to explain the assembly of the ecosystems of successional seres from species pools and abiotic environmental factors. The linguistic metaphor suggests that a theory of ecosystem assembly is a generative grammar that associates particular species with environmental features. The niche of a particular species is defined in terms of the rules of a generative-transformational grammar used to derive an hierarchically organized description of that species. This approach also has conceptual difficulties: (1) empirically testing competence theories and distinguishing possible and impossible objects (e.g. species and ecosystems) are difficult, (2) evolutionary processes are difficult to incorporate, (3) the mathematical tools associated with grammars do not have the range of applicability of the mathematical tools associated with the geometric metaphor, and (4) the productions (hierarchical trees) of a grammar are not easily visualized.The geometric and linguistic metaphors of the niche should be viewed as complementary pictures of the same object. The geometric representation is most adept at describing community dynamics and stability, niche overlap, and organism response to abiotic environments. The linguistic metaphor represents well organism activities, niche dynamics, and ecosystem assembly. (shrink)

This book introduces readers to the narrative structure of mathematical proofs and why mathematicians communicate that way, drawing examples from classic literature and employing metaphors and imagery.

The aim of the paper is to determine how metaphors tackle the probable nature of information and uncertainty in the structure of the communication process. Since the cognitive theory of conceptual metaphors holds that metaphoric thinking and doing are unavoidable, they are employed often in explaining the communicating domains. The metaphorical conceptualizing is recognized in Shannon and Weaver’s Mathematical Theory of Communication where such abstract concepts as freedom of choice, choosing probabilities, and uncertainty ware conceived in that (...) way. It is described in accord with Reddy’s conduit metaphor and Ritchie’s toolmakers paradigm. In the paper the issue of both the advantages and disadvantages of metaphors is considered: mainly, how they can explain and predict ways in which people communicate their expectations or uncertainties as well as, more practically, how the probable/informational metaphors enable the management of knowledge in libraries or databases. (shrink)

A formal model of metaphor is introduced. It models metaphor, first, as an interaction of “frames” according to the frame semantics, and then, as a wave function in Hilbert space. The practical way for a probability distribution and a corresponding wave function to be assigned to a given metaphor in a given language is considered. A series of formal definitions is deduced from this for: “representation”, “reality”, “language”, “ontology”, etc. All are based on Hilbert space. A few statements about a (...) quantum computer are implied: The sodefined reality is inherent and internal to it. It can report a result only “metaphorically”. It will demolish transmitting the result “literally”, i.e. absolutely exactly. A new and different formal definition of metaphor is introduced as a few entangled wave functions corresponding to different “signs” in different language formally defined as above. The change of frames as the change from the one to the other formal definition of metaphor is interpreted as a formal definition of thought. Four areas of cognition are unified as different but isomorphic interpretations of the mathematical model based on Hilbert space. These are: quantum mechanics, frame semantics, formal semantics by means of quantum computer, and the theory of metaphor in linguistics. (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

The purpose of this paper is to contextualize the study of metaphors within constructivist-informed research, in the hope that this process will orient cognitive scientists to the usefulness of implementing qualitative research methodologies, especially to using the person of the researcher as the primary research instrument. First, I explore some of the differences between Johnson and Lakoff’s Contemporary Metaphor Theory (CMT) and approaches evolving from it on one hand, and the clinical approach to metaphor based on a constructivist therapy (...) model, on the other. CMT has been one of the most significant forces that helped shift cognitive science toward an embodied approach to cognition. While it has succeeded to place physical experience back where it belongs in reason and meaning, CMT has, however, also fallen into some positivist traps which lead to problems such as a dualism, a split between the knower and the known, and with that, to a distrust of introspective, first-person accounts. In the process of finding conceptual metaphors — generalizations that govern metaphorical expressions — CMT often deletes the idiosyncratic characteristics and presuppositions implicit in linguistic metaphors; it divorces them from people’s sensory experiences, the “here and now” and the intent of their communications. The constructivist approach to metaphor that I present here accepts as a priori assumptions much of what workers of CMT are out to prove. In particular, it takes the correlation of conceptual metaphors and physical experience, as well as the unity of language and thought as pragmatic givens. Emulating the constructivist therapist’s approach to metaphors, I show how it is possible to deconstruct conceptual metaphors into minute sensory distinctions, using one’s own person as the main tool, for the purpose of helping people change their experiences in desired ways, at will. I illustrate this process by numerous examples from a wide field of applications, including mathematics education and psychotherapy. (shrink)

L’autobiographie intellectuelle du Discours de la méthode fait reposer la vocation philosophique de Descartes sur un événement, souvent associé à la découverte des « fondements de la science admirable », le 10 novembre 1619. L’article rapporte ici cet événement à l’injonction à « l’étude de soi » prescrite par les premières pages de la Sagesse de Charron. Il montre ensuite les similitudes entre la structure narrative du récit cartésien (le voyage, la halte, le choix du chemin, la mathesis comme « (...) chemin sûr ») et les auto-fictions narratives publiées par le mathématicien Caspar Grünewald entre 1617 et 1619. Enfin, il montre que le « désir d’apprendre à distinguer le vrai d’avec le faux » réécrit, sans lyrisme ni fioritures, le récit séminal du placard de la licence en droit (1616), où le récit de la vocation entrecroisait poétiquement les références mythologiques et théologiques. (shrink)

ABSTRACT Mathematicians frequently use aesthetic vocabulary and sometimes even describe themselves as engaged in producing art. Yet aestheticians, in so far as they have discussed this at all, have often downplayed the ascriptions of aesthetic properties as metaphorical. In this paper I argue firstly that the aesthetic talk should be taken literally, and secondly that it is at least reasonable to classify some mathematics as art.

Languages play an essential role in communicating aesthetic, scientific and religious convictions, as well as laws, worldviews and truths. Additionally, metaphors are an essential part of many languages and artistic expressions. In this paper I will first examine the role metaphors play in religion and art. Is there a specific focus on symbolic and metaphoric language in religion and art? Where are the analogies to be found in artistic metaphors and religious ones? How are differences to be (...) described? How do various concepts of aesthetics and theological concepts explain those different kinds of language and how, if at all, do they make use of them? Lastly: what could be added to aesthetics, philosophy and theology by examining carefully the role and importance of language, including nonverbal, sign language and especially metaphorical language? Without the human capacity for language, religions are scarcely imaginable. A widening of traditional exegesis and hermeneutics by taking into account nonverbal semantics is needed. Religion is a cognitive and linguistic phenomenon. By taking this seriously, we set and enable an agenda to discuss religion scientifically, leaving aside for the purpose of a scientific understanding and discourse about the inter-religious and the inner-religious claims of truth and absolutist claims. To sum it up: metaphor is introduced as an important means of language when it comes to religious conceptualization. Next, I will show that art, more than religion, deals with visual metaphor – the latter being an image that suggests a particular association, similarity or analogy between two generally unconnected visual elements. This often, but not always, functions in a roughly comparable fashion to the better-known concept of verbal metaphor. In addition, visual metaphor has developed many original and unique characteristics. These two sections are followed by another one dealing with cultural philosophy of religion and aesthetics, as well as the meaning of metaphors for these disciplines. The next section is on metaphor and metaphorical language in mathematics, natural sciences and art and how they are related, i.e. influence and help each other. I will discuss the critical approach to metaphors in natural science and provide a short introduction to the cultural history of mathematics and art. Mathematicians and artists have long been on the quest to understand the physical world they see before them and the abstract objects they know by thought alone. How have art and mathematics helped each other in representing each other’s concepts? A final section provides a summary and an outlook: theology is contextual as is science – and so is art. All these disciplines partly rely upon metaphor and by the help of metaphor get closer to an intercultural and interdisciplinary understanding. I shall argue that, by dealing more carefully with their metaphorical language and their own metaphors, together they become better equipped to map the world. (shrink)

The relation between theoretical models and metaphors has been studied since at least the 1950s. The relation between metaphors and mathematical modelling is less well researched. This article takes the media coverage of the foot and mouth modelling exercise in 2001 as an occasion to examine the metaphors of mathematical modelling that were proposed by the UK press during that time to make sense of this new scientific policy tool. One can detect a gradual change (...) in metaphor use by the newspapers from conceptualizing modellers as detectives and models as mapping tools to modellers as soldiers and heroes, to modellers as liars and models as tools to distort the truth. This seems to indicate a shift in reporting from seeing models as a legitimate and "objective" basis used by decision makers to pursue science-based policies towards seeing models as tools used to legitimize increasingly difficult political decisions. (shrink)

The terms complexity, complex adaptive systems, and sciences of complexity are found often in recent scientific literature, reflecting the remarkable growth in collaborative academic research focused on complexity from the origin and dynamics of organisms to the largest social and political organizations. One of the great challenges in this field of research is to discover which features are essential and shared by all of the seemingly disparate systems that are described as complex. Is there sufficient synthesis to suggest the possibility (...) of an overarching science of complexity? This report describes current views on this subject held by various eminent scholars associated with the Santa Fe Institute.The physical sciences have traditionally been concerned with “simple” systems whose dynamics can be described in mathematical terms with precision and certainty. In contrast, the biological and social sciences are inevitably concerned with self-organized or social “complex” systems whose detailed behaviors appear to be unpredictable. The two categories differ greatly in size and diversity, prompting the late mathematician Stanislaus Ulam to remark that research on complex systems might be compared to the study of non-elephants. Nevertheless, certain integrative themes have begun to emerge.Rising activity in this field of research runs completely counter to the trend toward increasing fragmentation and specialization in the sciences. It has stimulated a resurgence of interest in a broad synthesis involving mathematics, computational science, physics, chemistry, biology, neuroscience, and the social sciences. The growth of effort in this very extended field has been greatly stimulated by the development of new computational tools that are capable of dealing with vast, interrelated databases. Many of the participants in complexity research feel that it is now time to reintegrate the fragmented interests of much of the academic community. The reader is encouraged to consider whether such views are sparking a historic renaissance of scholarship or represent a passing scientific diversion. (shrink)

This chapter contains sections titled: A Lover's Question The Game of Love Where Did Our Love Go? Love is Strange.