Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...) the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

This paper argues that a theory of situated vision, suited for the dual purposes of object recognition and the control of action, will have to provide something more than a system that constructs a conceptual representation from visual stimuli: it will also need to provide a special kind of direct (preconceptual, unmediated) connection between elements of a visual representation and certain elements in the world. Like natural language demonstratives (such as `this' or `that') this direct connection allows entities to be (...) referred to without being categorized or conceptualized. Several reasons are given for why we need such a preconcep- tual mechanism which individuates and keeps track of several individual objects in the world. One is that early vision must pick out and compute the relation among several individual objects while ignoring their properties. Another is that incrementally computing and updating representations of a dynamic scene requires keeping track of token individuals despite changes in their properties or locations. It is then noted that a mechanism meeting these requirements has already been proposed in order to account for a number of disparate empiri- cal phenomena, including subitizing, search-subset selection and multiple object tracking (Pylyshyn et al., Canadian Journal of Experimental Psychology 48(2) (1994) 260). This mechanism, called a visual index or FINST, is brie. (shrink)

Philosophers from Hume, Kant, and Wittgenstein to the recent realists and antirealists have sought to answer the question, What are concepts? This book provides a detailed, systematic, and accessible introduction to an original philosophical theory of concepts that Christopher Peacocke has developed in recent years to explain facts about the nature of thought, including its systematic character, its relations to truth and reference, and its normative dimension. Particular concepts are also treated within the general framework: perceptual concepts, logical concepts, and (...) the concept of belief are discussed in detail. The general theory is further applied in answering the question of how the ontology of concepts can be of use in classifying mental states, and in discussing the proper relation between philosophical and psychological theories of concepts. Finally, the theory of concepts is used to motivate a nonverificationist theory of the limits of intelligible thought. Peacocke treats content as broad rather than narrow, and his account is nonreductive and non-Quinean. Yet Peacocke also argues for an interactive relationship between philosophical and psychological theories of concepts, and he plots many connections with work in cognitive psychology. (shrink)

One promising way to investigate the genealogy of norms is by considering not the origin of norms, but rather, what makes certain norms more likely to prevail. Emotional responses, I maintain, constitute one important set of mechanisms that affects the cultural viability of norms. To corroborate this, I exploit historical evidence indicating that 16th century etiquette norms prohibiting disgusting actions were much more likely to survive than other 16th century etiquette norms. This case suggests more broadly that work on cultural (...) evolution should pay greater attention to the role of emotion systems in cultural transmission. (shrink)

This article brings together two independent lines of research on numerally quantified expressions, e.g. two girls. One stems from work in linguistic theory and asks what truth conditional contributions such expressions make to the utterances in which they are used--in other words, what do numerals mean? The other comes from the study of language development and asks when and how children learn the meaning of such expressions. My goal is to show that when integrated, these two perspectives can both constrain (...) and enrich each other in ways hitherto not considered. Specifically, work in linguistic theory suggests that in addition to their 'exactly n' interpretation, numerally quantified NPs such as two hoops can also receive an 'at least n' and an 'at most n' interpretation, e.g. you need to put two hoops on the pole to win (i.e. at least two hoops) and you can miss two shots and still win (i.e. at most two shots). I demonstrate here through the results of three sets of experiments that by the age of 5 children have implicit knowledge of the fact that expressions like two N can be interpreted as 'at least two N' and 'at most two N' while they do not yet know the meaning of corresponding expressions such as at least/most two N which convey these senses explicitly. I show that these results have important implications for theories of the semantics of numerals and that they raise new questions for developmental accounts of the number vocabulary. (shrink)

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A.. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with a representational system that clearly (...) does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles. (shrink)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports (...) exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

Edmund Husserl is the founder of phenomenology. The Logical Investigations is Edmund Husserl's most famous work and has had a decisive impact on the direction of twentieth century philosophy. This is the first time both volumes of this classic work, translated by J.N. Findlay, have been available in paperback. They include a new introduction by Dermot Moran, placing the Logical Investigations in historical context and bringing out its importance for contemporary philosophy.

Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the question “What (...) is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

Reexamines the Western philosophical tradition, looking at the basic concepts of the mind, time, causation, morality, and the self.

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second Philosopher". (...) mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practice Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized. (shrink)

Lance Rips describes a unified theory of natural deductive reasoning and fashions a working model of deduction, with strong experimental support, that is capable of playing a central role in mental life.

Sigmund Freud, the founder of psychoanalysis, declared that religion is a universal obsessional neurosis in his famous work of 1927, The Future of an Illusion. This work provoked immediate controversy and has continued to be an important reference for anyone interested in the intersection of philosophy, psychology, religion, and culture. Included in this volume is Oskar Pfister's critical engagement with Freud's views on religion. Pfister, a Swiss pastor and lay analyst, defends mature religion from Freud's "scientism." Freud's and Pfister's texts (...) have been updated in Gregory C. Richter's translations from the original German. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...) of great interest to philosophers of language, linguists, metaphysicians, and logicians. (shrink)