An intervening gestural stage in language evolution, though seductive, is ultimately redundant, and is not necessarily supported by modern human or chimp behaviour. The findings and arguments offered from mirror neurones, anatomy, and lateralization are capable of other interpretations, and the manipulative dextrality of chimps is under-recognized. While language certainly possesses certain unique properties, its roots are ancient.

Keller & Miller (K&M) briefly mention and promptly dismiss the idea that genes for harmful mental disorders may confer certain advantages to affected individuals. However, the authors fail to consider that the same genes (in low doses or reduced penetrance) may be adaptive for relatives, and that this may in part explain why they are retained in the gene pool. (Published Online November 9 2006).

Adaptation to right-shifting prisms improves left neglect for mental number line bisection. This study examined whether adaptation affects the mental number line in normal participants. Thirty-six participants completed a mental number line task before and after adaptation to either: left-shifting prisms, right-shifting prisms or control spectacles that did not shift the visual scene. Participants viewed number triplets (e.g. 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Participants demonstrated (...) a leftward bias (i.e. overestimated the length occupied by numbers located on the left side of the number line) that was consistent with the effect of pseudoneglect. The leftward bias was corrected by a short period of visuomotor adaptation to left-shifting prisms, but remained unaffected by adaptation to right-shifting prisms and control spectacles. The findings demonstrate that a simple visuomotor task alters the representation of space on the mental number line in normal participants. (shrink)

While explaining a large proportion of any variance, accounts of the speed and accuracy of targetting movements use techniques (e.g., log transforms) that typically reduce variability before ''explaining'' the data. Therefore the predictive power of such accounts are important. We consider whether Plamondon's model can account for kinematics of targetting movements of clinical populations.

In lieu of an abstract, here is a brief excerpt of the content:"ALL EXISTING IS THE ACTION OF GOD": THE PHILOSOPHICAL THEOLOGY OF DAVID BRAINE DAVID BRADSHAW University ofTexas at Austin Austin, Texas Thou lovest all the things that are, and abhorrest nothing which thou hast made: for never wouldest thou have made any thing, if thou hadst hated il And how could any thing have endured, if it had not been thy will? or been preserved, if not called (...) by thee? But thou sparest all: for they are thine, 0 Lord, thou lover of souls. Wisdom 11:24-26 He is not far from every one of us; for in Him we live, and move, and have our being. Acts 17:27-28 I T HAS long been traditional to regard God as upholding all things in existence. This belief is known as the doctrine of continuous creation; it is a doctrine widely shared by Jews, Christians, and Moslems, and was first clearly articulated by St. Augustine.1 The phrase "continuous creation" can be misleading, for there is more to the doctrine than simply the extension of God's creative act through time. If that were all that is at stake, we would not need to revise our ordinary notion of the creator as craftsman or artist; we might think of God as like a painter who 1 Augustine, De Genesi ad Litteram iv.12. The history of the development of this doctrine has yet to be written, but important sources include Timaeus 4lab; Philo of Alexandria, De Providentia i.7; Colossians 1:17, Hebrews 1:3; John of Damascus, De Fide Orthodoxa i.3; Aquinas, Summa theologiae I, q. 104, a. 1. See also David Winston, Philo of Alexandria (New York: Paulist Press, 1981), 14-17; G. L. Prestige, God in Patristic Thought (London: S.P.C.K., 1964), 31-36; Richard Sorabji, Time, Creation,-and the Continuum (Ithaca: Cornell, 1983), 303-304. 379 380 DAVID BRADSHAW never stops adding new scenes, or a writer continually elaborating his story. That picture will not do. Ifthe painter were to cease painting, what he has made up to that point would remain in existence, whereas it is precisely the ability to stand on its own that the doctrine of continuous creation seeks to deny of what is made. Nor does anything in the relation of artisan to artifact properly convey the special intimacy and presence that the act of continuous creation is supposed to establish between God and creatures. A painting or a story does not "live and move and have its being" in the one who fashions it-nor would even a living and thinking creature, if the manner of its making were solely that of an artisan. These shortcomings raise the question of whether we should seek to replace the image of artisan and artifact with one that is more dynamic. We might think of God as like a singer and of what is made as His song, or of God as like a dancer and of what is made as His dance. By envisioning the act of creation as a "doing" rather than a "making," analogies of this type more adequately indicate the continuing ontological dependence ofcreation on Creator. They also draw our attention to a mode of presence radically different from that of one body to another, the presence of an agent "within" his own actions. They may thus bring us closer to what St. Paul presumably has in mind in the passage quoted from Acts. On the other hand, they have faults of their own, for they are hard to reconcile with creaturely freedom and the reality of secondary causation. How could any kind of "doing" act contrary to the will of its doer? Indeed, how could it act at all? Actions are not causal agents; a song does not act in its own right, but is at best an instrument of the one who sings it. So it would seem that, taken to their logical conclusion, analogies such as that of the singer and his song lead to the denial of both creatures' freedom and their causal efficacy. There have been thinkers in each of the religious traditions mentioned who have been willing to accept these... (shrink)

Language started evolving early, before gesture; commonalities of generativity between language and praxis have been over-emphasized. Language did not drive hominid brain evolution, evolving multifactorially and interactively. More than communication, it permits a cognitive modelling of reality and hierarchical data management. As the interactive sum of various cognitive and linguistic systems, and of brain structures each semi-independently evolved, it is not part of a single, general-purpose cognitive processor, nor is it a separately-evolved quasi-independent module.

To students and practitioners of anamorphic art, the name of Jean-François Niceron is more than preeminent; it has become iconic. La Perspective Curieuse was first published in 1638. An augmented version was then translated into Latin by Mersenne in 1646. A newly amended and augmented version was retranslated into French by Roberval in 1652. This book is an English translation of the 1652 text, with reference to the 1638 and 1646 versions. Considering the continued high reputation of the book, the (...) mathematics have been checked for correctness and consistency, and the authors have provided a full commentary, pointing out the most difficult turns of the 17th-century French, the respective contributions of Niceron, Mersenne and Roberval, and explaining Niceron's greatest insights and weaknesses. (shrink)

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.

John Pollock aims to construct a theory of rational decision making for real agents--not ideal agents. Pollock argues that theories of ideal rationality are largely irrelevant to the decision making of real agents. Thinking about Acting aims to provide a theory of "real rationality.".

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...) continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an important form of type theory (...) based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy students and professional philosophers who have learned some (...) classical first-order logic and would like to learn about other logics important to their philosophical work. (shrink)

5.5. Every topos is linguistic: the equivalence theorem.

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.

IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...) I have elected to offer detailed technical presentations of just a few important instances. (shrink)

We investigate Hilbert’s varepsilon -calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed varepsilon -terms. We extend the usual topos semantics for type theories to the varepsilon -operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined‘ varepsilon -term. MSC: 03B15, 03B20, 03G30.

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...) will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...) of transformation of structure (homomorphism) was central from the beginning, and it soon became apparent to algebraists that its most important concepts and constructions were in fact formulable in terms of that idea alone. Thus emerged the view that the essence of a mathematical structure is to be sought not in its internal constitution, but rather in the nature of its relationships with other structures of the same kind, as manifested through the network of transformations. This idea has achieved its fullest expression in category theory, an axiomatic framework within which the notions of transformation (as morphism or arrow) and composition (and also structure, as object) are fundamental, that is, are not defined in terms of anything else. (shrink)

Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In _Oppositions and Paradoxes_, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell (...) not only motivates abstract conceptual thinking about the paradoxes at issue, but he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life — he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or think of an office mailbox labelled “mail for those with no mailbox”—if this is a person’s mailbox, how can they possibly have “no mailbox”? These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts. (shrink)