The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the (...) proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially (...) shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the (...) realisation that, while arithmetic is a human creation, the development of arithmetic is constrained by our evolutionarily developed cognitive architecture. Arithmetic is a sophisticated cultural development, but it is ultimately based on abilities with numerosities that we already possess as infants and share with many non-human animals. Therefore, arithmetic is not purely conventional, an arbitrary game akin to chess. Instead, arithmetic is deeply connected to our basic cognitive capacities. (shrink)

Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist (...) view. I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way universal proto-arithmetical abilities determine the development of arithmetical cognition. (shrink)

One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I (...) will show that since the development of arithmetic is (partly) determined by biologically evolved proto-arithmetical abilities, arithmetical knowledge can be understood as maximally intersubjective. This maximal intersubjectivity, I argue, can lead to the experience of objectivity, thus providing a solution to the problem of reconciling non-platonist philosophy of mathematics with the (apparent) objectivity of mathematical knowledge. (shrink)

In the first part of the article, Giuseppe Veronese’s concept of arithmetical constructivism is reconstructed from his dispersed remarks. It is pointed out that although for Veronese time is a necessary condition for the construction of natural numbers by an individual subject and the subject cognizes time in an a priori way, it is not a (proto-)intuition of the subject. This is a fundamental difference between the concept proposed by Veronese and the constructivism of Kant and Brouwer. Veronese’s justification (...) of the subject’s ability to cognize infinity, represented in his text by infinite sequences, is analyzed in the second part of the paper. The only condition for the cognition of the infinite sequence is to have a rule according to which its successive terms “follow one another.” The price of the ability to cognize the infinitive “whole” (sequence) may be the uncognizability of certain “parts” (terms of sequence). Infinite sequences, cognizable according to Veronese, are allowed as objects of mathematical research, although they do not meet the condition of constructability. (shrink)

In the Poroi, Xenophon’s radical solution to Athens’ financial problems includes several ideas vital to the field of political economy. His identification of justice with the pursuit of wealth provides an alternative to the power politics that for half a century had taken Athens into a series of self-destructive imperial wars. He supports his idea of economic growth with arithmetic calculations, and he connects the results to traditional Greek views of public rewards and benefits. From this he crafts a (...) goal-directed strategy for economic growth designed to foster good will through incentives rather than coercion. This brief commentary on the text shows how Xenophon’s positive claims are based not on a modern demand-side conception of economic stimulation, but rather on building productive capital. He is a proto-Saysian, not a Keynesian. Xenophon’s proposals range beyond the polis into a pan-Hellenic vision of increasing trade that is centred on Athens, monetized with Athenian coinage, and idealized into the common peace that the Greeks had so long desired but so little achieved. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.1 (2003) 125-126 [Access article in PDF] Sextus Empiricus. Against the Grammarians (Adversos Mathematicos I). Introduction, Commentary, and Translation by D. L. Blank. Oxford: Clarendon Press, 1998. Pp. lvi + 436. Cloth, $105.00. Sesto Empirico. Contro gli astrologi. Introduction, Commentary, and Translation by Emidio Spinelli. Naples: Bibliopolis, 2000. Pp. 230. Paper, L. 70.000. No historian of philosophy should be retailing the old canards (...) that the Hellenistic philosopher Sextus Empiricus was silly, wholly confused, or merely a compiler and not a philosopher in his own right. These translations and commentaries, along with the recent work of Richard Bett, James Hankinson, and others, should bury those notions forever.Against the Grammarians is the first and Against the Astrologers is the fifth volume of Sextus's six-volume Against the Teachers of the technai or professional arts and sciences (grammar, rhetoric, geometry, arithmetic, astrology, and music). Sextus is not against all forms of the study of these matters; in fact, one of the skeptic's "rules for living" is that he will engage in an art (techne) to make a living (Outlines of Pyrrhonism, I.24). But he is against the claims of each of them to special technical expertise. He prefers the unpretentious experience of everyday life—living by the phenomena; non-expert, practical forms of grammar, astronomy, music—to fancy theories about the nature of things.Against the Grammarians is Sextus's critique of the interpreters of poetry and prose—today's English professors and historians. Philosophers will be amused by his deconstruction of various sorts of lit crit arrogance and rashness that are still with us. Regrettably, some of us are not immune from the same critiques.Against the Astrologers is Sextus's critique of the Chaldean soothsayers. He is not against the observation of celestial phenomena for agricultural or navigational purposes. His critiques of the astrologers for lack of empirical evidence and promotion of superstition make him sound like a proto-scientist.One originality of Sextus's work was his combination of a dogmatic Epicurean critique of the grammatologists and astrologers with a skeptical aporetic critique. Precisely because he does not hesitate to make use of the arguments of others, some interpreters have decided that Sextus was unoriginal and derivative. But Blank and Spinelli both show that he recombines these arguments in ways that no one else had done. That is one of the ways of being original in philosophy. The upshot is that he is our earliest source of quite a number of critiques of grammarians and astrologers.The use of dogmatic arguments raises the problem of whether Sextus engages in so-called "negative dogmatism" in these volumes. Right at the beginning of Against the Astrologers, Sextus accuses them of violating "right reason." What right does a skeptic have to invoke such a standard? Some have felt that at this stage Sextus was a dogmatic skeptic, invoking one set of truths to tear down another. But that is not necessary. Blank and Spinelli explain this well with a combination of points. One, we can see the dogmatic and Epicurean attacks as clearing the ground for the following skeptical equipollence of arguments. Two, we can follow Jonathan Barnes in assuming that each claim that something has been demolished assumes the counterargument that it has been proven, so that the implied upshot is equipollence. Three, "right reason" can be the sort of unpretentious practical reasoning and living in accord with the appearances that the skeptics often claimed to live by.Spinelli and Blank provide many useful improvements over the previous English translations of these texts by Bury and the Italian translations by Russo. Blank rightly relies on Spinelli's 1995 translation and commentary on Sextus's Against the Ethicists and Spinelli rightly relies on Blank for a number of points. Spinelli supplies the Greek on facing pages of the translation; Blank does not. Spinelli also supplies five Greek zodiac charts and a beautiful print of a fifteenth-century zodiac.Neither Blank nor Spinelli unduly lionizes Sextus. They admit that... (shrink)

In the Poroi, Xenophon's radical solution to Athens' financial problems includes several ideas vital to the field of political economy. His identification of justice with the pursuit of wealth provides an alternative to the power politics that for half a century had taken Athens into a series of self-destructive imperial wars. He supports his idea of economic growth with arithmetic calculations, and he connects the results to traditional Greek views of public rewards and benefits. From this he crafts a (...) goal-directed strategy for economic growth designed to foster good will through incentives rather than coercion. This brief commentary on the text shows how Xenophon's positive claims are based not on a modern demand-side conception of economic stimulation, but rather on building productive capital. He is a proto-Saysian, not a Keynesian. Xenophon's proposals range beyond the polis into a pan-Hellenic vision of increasing trade that is centred on Athens, monetized with Athenian coinage, and idealized into the common peace that the Greeks had so long desired but so little achieved. (shrink)

This chapter contains sections titled: Logical and Mathematical Background Intuitive versus Abstract Knowledge Logical Intuition and Abstract Reasoning Intuitive and Abstract Knowledge of Mathematics Principium Individuationis, Physics and Idealist Metaphysics of Space and Time Schopenhauer's Philosophical Geometry Intuitive Reduction of Arithmetic to Counting in Time Notes References Further Reading.

This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books (...) while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill1,2. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations3,4, and their performance suffers if this nonsymbolic system is impaired5. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is (...) required6–10. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction11–15. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics. We presented children with approximate symbolic arithmetic problems in a format that parallels previous tests of non-symbolic arithmetic in preschool children8,9. In the first experiment, five- to six-year-old children were given problems such as ‘‘If you had twenty-four stickers and I gave you twenty-seven more, would you have more or less than thirty-five stickers?’’. Children performed well above chance (65.0%, t1952.77, P 5 0.012) without resorting to guessing or comparison strategies that could serve as alternatives to arithmetic. Children who have been taught no symbolic arithmetic therefore have some ability to perform symbolic addition problems. The children’s performance nevertheless fell short of performance on non-symbolic arithmetic tasks using equivalent addition problems with numbers presented as arrays of dots and with the addition operation conveyed by successive motions of the dots into a box (71.3% correct, F1,345 4.26, P 5 0.047)8.. (shrink)

The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the (...) purely arithmetical content of a categorical conceptual analysis of the notion of natural number. The chapter explores its conceptual stability in light of the apparently destabilizing effect, through extensibility, of the phenomenon of incompleteness. It also offers heuristic and conceptual support for the viewpoint that Peano Arithmetic is the strongest natural first-order system for arithmetic, and that there is a sense in which it is complete with respect to purely arithmetical truth. (shrink)

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that it is trivial that relevant arithmetic is absolutely consistent, but classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under, I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly (...) the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under, I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment. (shrink)

Scholarly opinions vary on what language is, how it evolved, and from where or what it evolved. Long considered uniquely human, today scholars argue for evolutionary continuity between human language and animal communication systems. But while it is generally recognized that language is an evolving communication system, scholars continue to debate from which species language evolved, and what behavioral and cognitive features are the precursors to human language. To understand the nature of protolanguage, some look for homologs in gene functionality, (...) brain areas, or anatomical structures such as the supralaryngeal vocal tract; others point toward primates, their gestural, vocal, multimodal, and in later evolving hominins also their pantomimic communication systems; and still others draw parallels between the musicality that characterizes language and the pitch found in the numerous sounds produced by animals. Accordingly, protolanguage theories today are multiple and diverse, and protolanguages might have also been diverse. This special issue on Evolving (Proto)Language/s for Lingua bundles several of the protolanguage theories that were put forward at the sixth edition of the Ways to Protolanguage conference series, held at the Calouste Gulbenkian Foundation in Lisbon, in 2019. Not aimed at surveying all the different ways there are to conceptualize, study, and model protolanguage/s, this issue provides interested readers with good overviews on the role played in current theorizing on protolanguage/s by (paleo)anthropology, genetics, physiology, developmental, evolutionary, ecological, and pragmatic research lines. (shrink)

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial (...) simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

This paper connects the hard problem of consciousness to the interpretation of quantum mechanics. It shows that constitutive Russellian pan(proto)psychism (CRP) is compatible with Everett’s relative-state (RS) interpretation. Despite targeting different problems, CRP and RS are related, for they both establish symmetry between micro- and macrosystems, and both call for a deflationary account of Subject. The paper starts from formal arguments that demonstrate the incompatibility of CRP with alternative interpretations of quantum mechanics, followed by showing that RS entails Russellian (...) pan(proto)psychism. Therefore, CRP and RS are mutually supportive. It then provides a unified ontological picture by combining CRP and RS. The challenge faced by CRP, the combination problem, can be resolved by adopting a RS version of quantum mechanics. Technically, this is achieved by a co-consciousness relation capable of explaining the difference between first-person and third-person perspectives. The hierarchical structure of the relation removes any concern on the structural mismatch between the physical and the phenomenal. (shrink)

Inspired by Bermúdez’s notion of proto-logic, I would like to fathom what the true proto-logic could be like. But this will be approached only in a negative way of figuring out what it could not be. I shall argue that it could not be purely deductive by exploiting the recent researches in logic of maps. This will allow us to reorient the search for proto-logic, starting with animal abduction. I will also suggest that proto-logic won’t get (...) off the ground without proto-geometry. These negative results will shed some lights on some further conceptual and historical issues around the language of thought hypothesis to arrive at the true proto-logic. (shrink)

SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...) subject which begins when numerals are mentioned (not just used) and mentioned as names of numbers (not just as syntactic objects). Semantic arithmetic leads to many fascinating and surprising algorithms and decision procedures; it reveals in a vivid way the experiential import of mathematical propositions and the predictive power of mathematical knowledge; it provides an interesting perspective for philosophical, historical, and pedagogical studies of the growth of scientific knowledge and of the role metalinguistic discourse in scientific thought. (shrink)

This is an account of Sceptical investigation as it is presented by Sextus Empiricus. I focus attention on the motivation behind the Sceptic’s investigation, the goal of that investigation, and on the development Sextus describes from proto-Sceptical to Sceptical investigator. I suggest that recent accounts of the Sceptic’s investigative practice do not make sufficient sense of the fact that the Sceptic finds a relief from disturbance by way of suspending judgement, nor of the apparent continuity between proto-Sceptical and (...) Sceptical investigation. I offer an alternative account which turns on the suggestion that the Sceptic accepts that justification is the norm of belief. (shrink)

How is it that sounds from the mouth or marks on a page—which by themselves are nothing like things or events in the world—can be world-disclosive in an automatic manner? In this fascinating and important book, Lawrence J. Hatab presents a new vocabulary for Heidegger’s early phenomenology of being-in-the-world and applies it to the question of language. He takes language to be a mode of dwelling, in which there is an immediate, direct disclosure of meanings, and sketches an extensive picture (...) of proto-phenomenology, how it revises the posture of philosophy, and how this posture applies to the nature of language. Representational theories are not rejected but subordinated to a presentational account of immediate disclosure in concrete embodied life. The book critically addresses standard theories of language, such that typical questions in the philosophy of language are revised in a manner that avoids binary separations of language and world, speech and cognition, theory and practise, realism and idealism, internalism and externalism. (shrink)

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2-element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.

The proto-code of ethics and conduct for European nurse directors was developed as a strategic and dynamic document for nurse managers in Europe. It invites critical dialogue, reflective thinking about different situations, and the development of specific codes of ethics and conduct by nursing associations in different countries. The term proto-code is used for this document so that specifically country-orientated or organization-based and practical codes can be developed from it to guide professionals in more particular or situation-explicit reflection (...) and values. The proto-code of ethics and conduct for European nurse directors was designed and developed by the European Nurse Directors Association’s (ENDA) advisory team. This article gives short explanations of the code’ s preamble and two main parts: Nurse directors' ethical basis, and Principles of professional practice, which is divided into six specific points: competence, care, safety, staff, life-long learning and multi-sectorial working. (shrink)

The proto-code of ethics and conduct for European nurse directors was developed as a strategic and dynamic document for nurse managers in Europe. It invites critical dialogue, reflective thinking about different situations, and the development of specific codes of ethics and conduct by nursing associations in different countries. The term proto-code is used for this document so that specifically country-orientated or organization-based and practical codes can be developed from it to guide professionals in more particular or situation-explicit reflection (...) and values. The proto-code of ethics and conduct for European nurse directors was designed and developed by the European Nurse Directors Association’s (ENDA) advisory team. This article gives short explanations of the code’ s preamble and two main parts: Nurse directors' ethical basis, and Principles of professional practice, which is divided into six specific points: competence, care, safety, staff, life-long learning and multi-sectorial working. (shrink)

_The Foundations of Arithmetic_ is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It (...) was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (shrink)

What is it for a predicate or a general term to be a rigid designator? Two strategies for answering this question can be found in the literature, but both run into severe difficulties. In this paper, it is suggested that proper names and the usual examples of rigid predicates share a semantic feature which does the theoretical work usually attributed to rigidity. This feature cannot be equated with rigidity, but in the case of singular terms this feature entails their rigidity, (...) as understood in the standard characterisation. Hence, it is appropriate to call this feature proto-rigidity. (shrink)

This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of "non-existing" objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non (...) unlike the notion of "truth at world" found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization). (shrink)

... as 'logicism') that the content expressed by true propositions of arithmetic and analysis is not something of an irreducibly mathematical character, ...

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the (...) first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time. The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. (shrink)

This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...) mathematics and transcendental philosophy. (shrink)

This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of “non-existing” objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non (...) unlike the notion of “truth at world” found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization). (shrink)

How do we understand other individuals’ actions? Answers to this question cluster around two extremes: either by ascribing to the observed individual mental states such as intentions, or without ascribing any mental states. Thus, action understanding is either full-blown mindreading, or not mindreading. An intermediate option is lacking, but would be desirable for interpreting some experimental findings. I provide this intermediate option: actions may be understood by ascribing to the observed individual proto-intentions. Unlike intentions, proto-intentions are subject to (...) context-bound normative constraints, therefore being more widely available across development. Action understanding, when it consists in proto-intention ascription, can be a minimal form of mindreading. (shrink)

Through his innovative study of language, noted Heidegger scholar Lawrence Hatab offers a proto-phenomenological account of the lived world, the “first” world of factical life, where pre-reflective, immediate disclosiveness precedes and makes possible representational models of language. Common distinctions between mind and world, fact and value, cognition and affect miss the meaning-laden dimension of embodied, practical existence, where language and life are a matter of “dwelling in speech.” In this second volume, Hatab supplements and fortifies his initial analysis by (...) offering a detailed treatment of child development and language acquisition, which exhibit a proto-phenomenological world in the making. He then takes up an in-depth study of the differences between oral and written language (particularly in the ancient Greek world) and how the history of alphabetic literacy shows why Western philosophy came to emphasize objective, representational models of cognition and language, which conceal and pass over the presentational domain of dwelling in speech. Such a study offers significant new angles on the nature of philosophy and language. (shrink)

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity (...) and Associativity of Addition and Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

In this paper some of the history of the development of arithmetic in set theory is traced, particularly with reference to the problem of avoiding the assumption of an infinite set. Although the standard method of singling out a sequence of sets to be the natural numbers goes back to Zermelo, its development was more tortuous than is generally believed. We consider the development in the light of three desiderata for a solution and argue that they can probably not (...) all be satisfied simultaneously. (shrink)