In his celebrated paper of 1931 [7], Kurt Gödel proved the existence of sentences undecidable in the axiomatized theory of numbers. Gödel's proof is constructive and such a sentence may actually be written out. Of course, if we follow Gödel's original procedure the formula will be of enormous length.Forty-five years have passed since the appearance of Gödel's pioneering work. During this time enormous progress has been made in mathematical logic and recursive function theory. Many different mathematical problems have been proved (...) recursively unsolvable. Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate. We have only to carry out a suitable arithmetization. Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiency.In this article we construct an explicit undecidable arithmetical formula,F(x, n), in prenex normal form. The formula is explicit in the sense that it is written out in its entirety with no abbreviations. The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values ofn, whether or notF(n, n)is true or false. MoreoverF(n, n)is undecidable in the formal (axiomatic) sense of Gödel [7]. Given any of the usual axiomatic theories to which Gödel's Incompleteness Theorem applies, there exists a value ofnsuch thatF(n, n)is unprovable and irrefutable. Thus Gödel's Incompleteness Theorem can be “focused” into the formulaF(n, n). Thus some substitution instance ofF(n, n)is undecidable in Peano arithmetic, ZF set theory, etc. (shrink)

Shepard claims that “evolutionary internalization of universal regularities in the world” takes place. His position is interesting and seems plausible with regard to “default” motion detection and aspects of colour constancy which he addresses. However, his claim is not convincing with regard to object recognition. [Shepard].

A sentence like every circle is blue might be understood in terms of individuals and their properties or in terms of a relation between groups. Relatedly, theorists can specify the contents of universally quantified sentences in first-order or second-order terms. We offer new evidence that this logical first-order vs. second-order distinction corresponds to a psychologically robust individual vs. group distinction that has behavioral repercussions. Participants were shown displays of dots and asked to evaluate sentences with each, every, or all combined (...) with a predicate. We find that participants are better at estimating how many things the predicate applied to after evaluating sentences in which universal quantification is indicated with every or all, as opposed to each. We argue that every and all are understood in second-order terms that encourage group representation, while each is understood in first-order terms that encourage individual representation. Since the sentences that participants evaluate are truth-conditionally equivalent, our results also bear on questions concerning how meanings are related to truth-conditions. (shrink)

The paper raises the question of the function of visual representations in medieval cosmographical texts. It proposes to view diverse functions of figures in relation to changing discursive environments, including differing philosophical positions and changing social and intellectual contexts. It further suggests a distinction between figures that were elaborated within the highly specialized disciplines of mathematics and philosophy of nature in Greek Antiquity and figures that were instrumental in transmitting accepted world models, thus avoiding the opposition between scientific and unscientific (...) types of verbal and pictorial documents. Simplifying changes, when figures are abstracted form their geometrical context and accompany doxographical, descriptive accounts, are characterized in terms of schematization. Concomitantly, mathematical and philosophical demonstrations tend to give way to proofs of a predominantly rhetorical nature: images are verbally construed and, in order to enhance these, actual visual figures— mostly linear, diagrammatic constructs—are added. With regard to the Middle Ages, the paper distinguishes two principal periods: the period from the seventh to the eleventh century and the period of the so-called twelfth-century Renaissance. First, the verbal and pictorial cosmological corpus of Roman origin gave rise to explanations and variations but not to consequential theoretical developments and cosmological diagrams tended to fuse with summarizing tables at this time. Then, during the twelfth century, mathematical and philosophical documents of a specialized kind that were translated from the Arabic and also from the Greek became available in the Latin West. In mathematics, specialized types of study remained, however, sparse. Continuous elaborations of the assimilated material set in later only, within the thirteenth-century university context. Nevertheless, twelfth-century authors of cosmographical accounts became increasingly aware that their expositions and visual figures were ultimately derived from geometrical models of the universe. More diversified types of demonstration and corresponding visual figures were being used, as exemplified by William of Conches’ Dragmaticon philosophiae. (shrink)

A central component of the cognitive revolution is a commitment to explaining behaviour by reference to internal representations of the world. This core aspect.

The objectivity of representation Content Type Journal Article DOI 10.1007/s11016-010-9505-1 Authors Alfredo Paternoster, University of Bergamo, Via Pignolo 123, 24121 Bergamo, Italy Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

As an object which is new, complex and potentially challenging some of the foundations of the Social Order (SO), we sought to study the influence of the Political Orientation (PO) and Social Representations (SR) of the Social Order (Staerklé et al., 2007). Qui a droit à quoi? Représentations et légitimation de l’ordre social. PUG) on the socio-representational construction of the Universal Basic Income (UBI) and stances towards this measure (attitude and estimated fair amount). Data were collected via Facebook from 264 (...) French general population participants (Mage = 37.8; SD = 12.7; 174 women). Our Structural Equation Model revealed that the PO would be associated with the stances towards the UBI, mediated by the different ways of representing it and the representations of the SO. Results show two differents socio-representational logics in relation to being right-wing and one in relation to being left-wing. This leads to contrasting stances towards the UBI and seems to reflect different desires to maintain or change the SO. (shrink)

This paper argues in the first place that nominalists are right in insisting against ontological realists that semantic universality does not require commitment to universal entities. However, Ockham, in his zeal to get rid of Scotus’s universal entities, swept under the carpet the issue of universal representational content of genuinely universal symbols, conflating it with the mere indifference of the information content of non-distinctive singular representations. Buridan did come up with an abstractionist theory of the formation of genuinely universal representational (...) content to solve the resulting issues, however, the paper argues further, his solution is committed to attributing the sort of “aspectuality” to universal absolute concepts that his Ockhamist semantics denies to them. The conclusion of the paper suggests how Aquinas’s “moderate realism” can provide a consistent solution without the ontological extravaganza of ontological realists, without conflating the mere indifference of singular representation with genuine universality, and without having to deny aspectuality to our quidditative universal concepts formed by abstraction. (shrink)

This article aims to analyze the manifestation of the universal axiological cultural category of sinfulness in the media space. The study's subject was this category's properties and functions, which are actualized when represented in digital media. In accordance with this goal, we used such methods as continuous sampling and observation of the facts of the representation of cultural categories, which allowed us to determine the linguistic material for analysis and a statistical method for quantitative analysis. The content analysis of (...) the units selected for the study was used to identify the role, properties, and main functions of the universal axiological category of sinfulness. The novelty of the research lies in the fact that, for the first time, the universal nature of the category of sinfulness, its functions and properties, represented in media publications, were considered. Main conclusions: the study of the facts of actualization of the category of sinfulness in the media space allowed us to assert that the phenomenon under study is associated with topics such as politics, crime (in news media), and the assessment of citizens' behavior and morality (in religious media). These facts indicate that the primary function of the category of sinfulness is regulatory, aimed at evaluating and correcting the actions of members of society, as well as creating public opinion. In addition, the article proves that the category of sinfulness has transcultural properties, penetrating and being realized in all cultural layers, from local to global. In these cultural layers, the nature of the category acquires a different shade. In particular, at the global level, the category of sinfulness acquires the meaning of guilt, which is a consequence of the significant influence of Western culture on the global culture. It seems that the revealed properties of the category indicate its high importance for modern Russian society. The established regulatory function will make it possible to apply this category, including within the framework of regulating and managing public opinion and promoting basic values. (shrink)

What does Jacques Derrida have to say about sport? Sport philosophers haven’t been particularly drawn to Derrida as a source of enlightenment in sport. Indeed, deconstructionist philosophy in gener...

Book review of Emily Grosholz's Representation and Productive Ambiguity in Mathematics and the Sciences (2007).

Why does female under- representation emerge during undergraduate education? At the University of Sydney, we surveyed students before and after their first philosophy course. We failed to find any evidence that this course disproportionately discouraged female students from continuing in philosophy relative to male students. Instead, we found evidence of an interaction effect between gender and existing attitudes about philosophy coming into tertiary education that appears at least partially responsible for this poor retention. At the first lecture, disproportionately (...) few female students intended to major. Further, at the first lecture, female students were less interested in philosophy, were less self-confident about philosophy, and were less able to imagine themselves as philosophers. Similarly, female students predicted they would feel more uncomfortable in philosophy classes than male students did. Further study with a control is warranted to determine whether this interaction effect is peculiar to philosophy, or whether it is indicative of a more general gendered trend amongst first year undergraduate students. (shrink)

Although recent empirical research suggests that there is a gender gap in Anglophone philosophy, no research has been done on the representation of women in non‐Anglophone philosophy. The present study constitutes a first step toward filling this void in the literature by providing empirical evidence on the representation of female students and female faculty members in Greek universities' departments of philosophy. Our findings indicate that the underrepresentation of female students in philosophy is not a universal phenomenon, since female (...) students constitute the majority of philosophy students in Greece at both the undergraduate and the graduate levels. However, our findings also suggest that the low number of women in philosophy at the faculty level is not a problem unique to Anglophone philosophy, since female faculty members comprise, on average, only 29% of philosophy faculty members in Greece. In order to explain these findings, we argue, first, that the teaching of philosophy at the secondary level may motivate female students in Greece to enter and persist in philosophy, and, second, that since the gender gap at the faculty level in Greece cannot be attributed to the low number of female students in the philosophy pipeline, the causes of women's poor participation in philosophy at the faculty level should be looked for elsewhere. (shrink)

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations—whatever they are—must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may (...) give rise to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an “assembly language” for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems. (shrink)

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

Scientific representation: A long journey from pragmatics to pragmatics Content Type Journal Article DOI 10.1007/s11016-010-9465-5 Authors James Ladyman, Department of Philosophy, University of Bristol, 9 Woodland Rd, Bristol, BS8 1TB UK Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA Mauricio Suárez, Department of Logic and Philosophy of Science, Complutense University of Madrid, 28040 Madrid, Spain Bas C. van Fraassen, Philosophy Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA (...) 94132, USA Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796. (shrink)

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether (...) an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. (shrink)

It is almost universally acknowledged that first-order logic (FOL), with its clean, well-understood syntax and semantics, allows for the clear expression of philosophical arguments and ideas. Indeed, an argument or philosophical theory rendered in FOL is perhaps the cleanest example there is of “representing philosophy”. A number of prominent syntactic and semantic properties of FOL reflect metaphysical presuppositions that stem from its Fregean origins, particularly the idea of an inviolable divide between concept and object. These presuppositions, taken at face value, (...) reflect a significant metaphysical viewpoint, one that can in fact hinder or prejudice the representation of philosophical ideas and arguments. Philosophers have of course noticed this and have, accordingly, sought to alter or extend traditional FOL in novel ways to reflect a more flexible and egalitarian metaphysical standpoint. The purpose of this paper, however, is to document and discuss how similar “adaptations” to FOL—culminating in a standardized framework known as Common Logic —have evolved out of the more practical and applied encounter of FOL with the problem of representing, sharing, and reasoning upon information on the World Wide Web. (shrink)

This book is nominally about linguistic representation. But, since it is we who do the representing, it is also about us. And, since it is the universe which we represent, it is also about the universe. In the end, then, this book is about everything, which, since it is a philosophy book, is as it should be. I recognize that it is nowadays unfashionable to write books about every thing. Philosophers of language, it will be said, ought to stick (...) to writing about language; philosophers of science, to writing about science; epis temologists, to writing about knowing; and so on. The real world, however, perversely refuses to carve itself up so neatly, and, although I recognize that the real w,orld is nowadays also unfashionable, in the end I judged that one might get closer to the truth of various matters by going along with it. So I have done so. lt was Wilfrid Sellars who initially convinced me of the virtues of this way of proceeding. At this point one normally says something like "The debt that this book owes him is immense". I would say it too, were it not to understate the case, From Wilfrid, I learned to think about things. If the upshot of my thinking tends, as it obviously does, to show a general con silience with the upshot of his, it is primarily because he is so very good at it - and he had a head start. (shrink)

This paper evaluates a general argument for the conclusion that visual representations in science must play the role of truth bearers if they are to figure as legitimate contributors to scientific arguments and explanations. The argument is found to be unsound. An alternative approach to assessing the role of visual representations in science is exemplified by an examination of the role of structural formulas in organic chemistry. Structural formulas are found not to play the role of truth bearers; nonetheless, they (...) contribute to the arguments and explanations of organic chemistry. An early success of conformational analysis is presented in order to illustrate the role of structural formulas in the discourse of organic chemistry. *Received January 2007; revised July 2009. †To contact the author, please write to: Department of Philosophy and Religious Studies, Rowan University, 201 Mullica Hill Road, Glassboro, NJ 08028; e‐mail: [email protected]. (shrink)

I argue against theories that attempt to reduce scientific representation to similarity or isomorphism. These reductive theories aim to radically naturalize the notion of representation, since they treat scientist's purposes and intentions as non-essential to representation. I distinguish between the means and the constituents of representation, and I argue that similarity and isomorphism are common but not universal means of representation. I then present four other arguments to show that similarity and isomorphism are not the (...) constituents of scientific representation. I finish by looking at the prospects for weakened versions of these theories, and I argue that only those that abandon the aim to radically naturalize scientific representation are likely to be successful. (shrink)

Universal Grammar (UG) is a theory about the innate linguistic knowledge that child language learners bring to the task of language acquisition. This chapter examines the findings of experimental research on children's knowledge of one principle of UG, called Principle C. It presents the defining properties of Principle C. The chapter reviews empirical evidence showing that children apply Principle C to a range of disparate‐looking phenomena. It also presents empirical findings that document children's assignment of hierarchical structure to strings of (...) words. The empirical evidence supports a conclusion – that child language learners access a level of linguistic representation in addition to the constituent structure that is formed using Merge. This additional component of the language apparatus is called Logical Form. The chapter looks at a string of words where Principle C applies at the level of Logical Form. Cleft sentences are one such case. (shrink)

Contents: Preface. Introduction. J. ECHEVERRIA, A. IBARRA and T. MORMANN: The Long and Winding Road to the Philosophy of Science in Spain. REPRESENTATION AND MEASUREMENT. A. IBARRA and T. MORMANN: Theories as Representations. J. GARRIDO GARRIDO: The Justification of Measurement. O. FERNÁNDEZ PRAT and D. QUESADA: Spatial Representations and Their Physical Content. J.A. DIEZ CALZADA: The Theory-Net of Interval Measurement Theory. TRUTH, RATIONALITY, AND METHOD. J.C. GARCÍA-BERMEJO OCHOA: Realism and Truth Approximation in Economic Theory. W.J. GONZALEZ: Rationality in Economics (...) and Scientific Predictions: A Critical Reconstruction of Bounded Rationality and Its Role in Economic Predictions. J.P. ZAMORA BONILLA: An Invitation to Methodonomics. LOGICS, SEMANTICS, AND THEORETICAL STRUCTURES. J.L. FALGUERA: A Basis for a Formal Semantics of Linguistic Formulations of Science. A. SOBRINO and E. TRILLAS: Can Fuzzy Logic Help to Pose Some Problems in the Philosophy of Science? J. de LORENZO: Demonstrative Ways in Mathematical Doing. M. CASANUEVA: Genetics and Fertilization: A Good Marriage. C.U. MOULINES: The Concept of Universe from a Metatheoretical Point of View. List of Contributors. Index of Names. (shrink)