Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

With the help of the natural history of “zero,” and the use of “zero” as a starting point, one may consider two types of metaphysics. On the one hand, the epistemological metaphysics, based on the perceptual/rational dichotomy, is related to the zero as a vacancy between numbers. On the other hand, the genetic metaphysics, based on the dichotomy of source-evolution (or origin and derivate), has much to do with the zero as a number between negative and positive numbers. In this (...) respect, zero represents the horizon of metaphysics: we can forever approach it, but we cannot ultimately arrive at it. Though serving as the point of convergence and divergence for all relationships, zero has no definable content of its own. Such is the essence of zero, and of metaphysics as well. (shrink)

Descartes’ Method of Radical Doubt was not radical enough. –A. Marcel (2003, 181) In short, heterophenomenology is nothing new; it is nothing other than the method that has been used by psychophysicists, cognitive psychologists, clinical neuropsychologists, and just about everybody who has ever purported to study human consciousness in a serious, scientific way. –D. Dennett (2003, 22).

EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)

I argue that uniquely human forms of ‘Theory of Mind’ are a product of cultural evolution. Specifically, propositional attitude psychology is a linguistically constructed folk model of the human mind, invented by our ancestors for a range of tasks and refined over successive generations of users. The construction of these folk models gave humans new tools for thinking and reasoning about mental states—and so imbued us with abilities not shared by non-linguistic species. I also argue that uniquely human forms of (...) ToM are not required for language development, such that an account of the cultural origins of ToM does not jeopardise the explanation of language development. Finally, I sketch a historical model of the cultural evolution of mental state talk. (shrink)

According to a chorus of authors, the human life-world is currently invaded by an avalanche of high-tech devices referred to as “emerging,” ”intimate,” or ”NBIC” technologies: a new type of contrivances or gadgets designed to optimize cognitive or sensory performance and / or to enable mood management. Rather than manipulating objects in the outside world, they are designed to influence human bodies and brains more directly, and on a molecular scale. In this paper, these devices will be framed as ‘extimate’ (...) technologies (both intimate and external; both embedded and foreign; both life-enhancing and intrusive), a concept borrowed from Jacques Lacan. Although Lacan is not commonly regarded as a philosopher of technology, the dialectical relationship between human desire and technological artefacts runs as an important thread through his work. Moreover, he was remarkably prescient concerning the blending of life science and computer science, which is such a distinctive feature of the current techno-scientific turn. Building on a series of Lacanian concepts, my aim is to develop a psychoanalytical diagnostic of the technological present. Finally, I will indicate how such an analysis may inform our understanding of human life and embodiment as such. (shrink)

The notion of a ‘symbol’ plays an important role in the disciplines of Philosophy, Psychology, Computer Science, and Cognitive Science. However, there is comparatively little agreement on how this notion is to be understood, either between disciplines, or even within particular disciplines. This paper does not attempt to defend some putatively ‘correct’ version of the concept of a ‘symbol.’ Rather, some terminological conventions are suggested, some constraints are proposed and a taxonomy of the kinds of issue that give rise to (...) disagreement is articulated. The goal here is to provide something like a ‘geography’ of the various notions of ‘symbol’ that have appeared in the various literatures, so as to highlight the key issues and to permit the focusing of attention upon the important dimensions. In particular, the relationship between ‘tokens’ and ‘symbols’ is addressed. The issue of designation is discussed in some detail. The distinction between simple and complex symbols is clarified and an apparently necessary condition for a system to be potentially symbol, or token bearing, is introduced. (shrink)

A defining characteristic of modern science is its ability to make immensely successful predictions of natural phenomena without invoking a putative god or a supernatural being. Here, we argue that this intellectual discipline would not acquire such an ability without the mathematical zero. We insist that zero and its basic operations were likely conceived in India based on a philosophy of nothing, and classify nothing into four categories—balance, absence, emptiness and nonexistence. We argue that zero is a tangible representation of (...) nonexistence and constitutes all nonzero numbers, which together represent existence. It appears that zero’s journey out of India somewhat separated its mathematical and philosophical aspects, with the former being more valued by some cultures and the latter by others. The European culture, in which modern science grew, largely ignores a philosophy of nothing due to a deep-rooted Greek philosophical base, although this science relies on the notion of nonexistence through zero. Consequently, zero is a mere number of convenience without its foundational philosophy in science, and techniques to circumvent zero are developed. We insist that, while such techniques contribute to the progress of science and mathematics within the current framework, a tendency to avoid zero and its philosophy leads to approximations and may hinder a deeper understanding. Finally, we argue that nonexistence may notionally constitute existence, and hence may be the fundamental. This implies that, if a supernatural being exists, it is not the fundamental. The independence of modern science from a supernatural being is consistent with this. (shrink)

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, (...) notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

Tanto nos Pensamentos quanto em seus trabalhos matemáticos, Pascal faz referência ao “nada”, assim como a um processo que poderíamos chamar de “aniquilamento”, segundo o qual aquilo que é finito se torna um nada diante do infinito. O “nada” pascaliano, segundo a interpretação aqui defendida, pode ter, em diferentes passagens da obra do autor, uma acepção relativa ou uma acepção absoluta, o que vale também para os termos de “infinito”, “desproporção” e “indivisível” na obra de Pascal. Além do valor de (...) tal análise para os Pensamentos, ela se propõe mostrar certa proximidade estrutural entre as obras matemáticas e apologéticas de Pascal. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

The possibilities are explored of considering nothing as the intended object of thoughts that are literally about the concept of nothing first, and thereby of nothing. Nothing, on the proposed analysis, turns out to be nothing other than the property of being an intendable object. There are propositions that look to be both true and to be about nothing in the sense of being about the concept and ultimate intended object of what is here formally defined and designated as N-nothing. (...) We have been thinking about it already in reading and understanding the meaning of this abstract. Nothing nothings, we shall assert in all seriousness and with a definite formally definable literal meaning. The concept of N-nothing in that sense is a concept with identity conditions like that of any entity, the difference being that the concept of N-nothing is a nonentity and nonexistent intended object. (shrink)

Computation and formalization are not modalities of pure abstractive operations. The essay tries to revise the assumption of the constitutive nonsensuality of the formal. The argument is that formalization is a kind of linear spatialization, which has significant visual dimensions. Thus, a connection can be discovered between visualization by figurative graphism and formalization by symbolic calculations: Both use spatial relations not only to represent but also to operate on epistemic, nonspatial, nonvisual entities. Descartes was one of the pioneers of using (...) this kind of two-dimensional spatiality as a cognitive instrument. (shrink)

Complex superpositions of degenerate hydrogen wavefunctions for the n th energy level can possess zero lines (phase singularities) in the form of knots and links. A recipe is given for constructing any torus knot. The simplest cases are constructed explicitly: the elementary link, requiring n≥6, and the trefoil knot, requiring n≥7. The knots are threaded by multistranded twisted chains of zeros. Some speculations about knots in general complex quantum energy eigenfunctions are presented.