The nature of changes in mathematics was discussed recently in Revolutions in Mathematics. The discussion was dominated by historical and sociological arguments. An obstacle to a philosophical analysis of this question lies in a discrepancy between our approach to formulas and to pictures. While formulas are understood as constituents of mathematical theories, pictures are viewed only as heuristic tools. Our idea is to consider the pictures contained in mathematical text, as expressions of a specific language. Thus we (...) get formulas and pictures into one linguistic framework, and so we are able to analyse their interplay in the course of history. (shrink)

In this, his magnum opus, distinguished linguist Zellig Harris presents a formal theory of language structure, in which syntax is characterized as an orderly system of departures from random combinations of sounds, words, and indeed of all elements of language.

Science has become, as all nonspecialists know to their cost, increasingly mathematical; science textbooks and research papers, even popularising articles in Scientific American, are littered with graphs, numbers, mathematical symbols and equations. This has prompted the question “What exactly is the function of mathematics in science?” For example, could one understand a theory such as Einstein’s theory of special relativity without having knowledge of any sophisticated mathematics?

For parents wanting their children to get a head start in reading, it can be a challenge to find something that will maintain their attention. Now, learning to read can become a fun and enter- taining thing to do with the help of an extraordinary cat. Join Cleo-cat-tra as she brings reading to life in the charming picture book Rhymes and Times with Cleo-cat-tra by Lucy T. Geringer and illustrated by Bernardita Cox Kollock. Rhymes and Times of Cleo-cat-tra is a (...) picture book about a special cat named Cleo-cat-tra. She is not an ordinary cat, as she cleverly rhymes about animals, people, and things she encounters inside and outside her house. The rhymes are silly which make learning a fun and creative pro- cess for kids and for parents to share. This delightful story makes use of colorful illustrations to heighten a child's imagination. Its simple text and brief sentences make reading easy for the early reader. It is also a great book for children who are learning English and can be enjoyed both at home and in school through lively interactive discussions between students and teachers. It is a book for all ages. (shrink)

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently. Like any designed system, it can perform well or poorly, and the philosophy of mathematics has a role (...) to play in helping us understand the general principles by which it serves its purposes well. (shrink)

Recent scholarship has helped to demythologise the life and work of Georg Philipp Friedrich von Hardenberg who, as the poet “Novalis”, had come to instantiate the nineteenth-century’s stereotype of the romantic poet. Among Hardenberg’s interests that seem to sit uneasily with this literary persona were his interests in science and mathematics, and especially in the idea, traceable back to Leibniz, of a mathematically based computational approach to language. Hardenberg’s approach to language, and his attempts to bring (...) class='Hi'>mathematics to bear on poetry, is examined in relation to debates that developed late in the eighteenth century over the relation of language to thought—debates which share many features with contemporary ones in this area. (shrink)

I discuss ways in which the linguistic form of mathimatics helps us think mathematically.

This special issue is a result of The Bucharest Colloquium in Analytic Philosophy dedicated to Frege's philosophy of mathematics and language. It was held at the Research Center for Logic, H...

The papers collected in this volume exemplify some of the trends in current approaches to logic, language and computation. Written by authors with varied academic backgrounds, the contributions are intended for an interdisciplinary audience. The first part of this volume addresses issues relevant for multi-agent systems: reasoning with incomplete information, reasoning about knowledge and beliefs, and reasoning about games. Proofs as formal objects form the subject of Part II. Topics covered include: contributions on logical frameworks, linear logic, and different (...) approaches to formalized reasoning. Part III focuses on representations and formal methods in linguistic theory, addressing the areas of comparative and temporal expressions, modal subordination, and compositionality. (shrink)

Modal logic is the study of modalities - expressions that qualify assertions about the truth of statements - like the ordinary language phrases necessarily, possibly, it is known/believed/ought to be, etc., and computationally or mathematically motivated expressions like provably, at the next state, or after the computation terminates. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades, since the introduction of the methods of Kripke semantics, and now impacts on a (...) wide range of disciplines, including the philosophy of language and linguistics ('possible words' semantics for natural language), constructive mathematics (intuitionistic logic), theoretical computer science (dynamic logic, temporal and other logics for concurrency), and category theory (sheaf semantics). This volume collects together a number of the author's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic modals, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Another paper on the 'Henkin method' in completeness proofs has been substantially extended to give new applications. Additional articles are concerned with quantum logic, provability logic, the temporal logic of relativistic spacetime, modalities in topos theory, and the logic of programs. (shrink)

In this book one of the world's foremost philosophers of language presents his unifying vision of the field--its principal achievements, its most pressing current questions, and its most promising future directions. In addition to explaining the progress philosophers have made toward creating a theoretical framework for the study of language, Scott Soames investigates foundational concepts--such as truth, reference, and meaning--that are central to the philosophy of language and important to philosophy as a whole. The first part of (...) the book describes how philosophers from Frege, Russell, Tarski, and Carnap to Kripke, Kaplan, and Montague developed precise techniques for understanding the languages of logic and mathematics, and how these techniques have been refined and extended to the study of natural human languages. The book then builds on this account, exploring new thinking about propositions, possibility, and the relationship between meaning, assertion, and other aspects of language use. An invaluable overview of the philosophy of language by one of its most important practitioners, this book will be essential reading for all serious students of philosophy. (shrink)

Four modes of language acquisition and communication are presented translating ancient Indian expressions on human consciousness, mind, their form, structure and function clubbing with the Sabdabrahma theory of language acquisition and communication. The modern scientific understanding of such an insight is discussed. . A flowchart of language processing in humans will be given. A gross model of human language acquisition, comprehension and communication process forming the basis to develop software for relevantmind-machine modeling will be presented. The (...) implications of such a model to artificial intelligence and cognitive sciences will be discussed. The essential nature and necessity of a physics, communication engineering , biophysical and biochemical insight as both complementary and supplementary to using mathematical and computational methods in delineating the theory of language processing in humans is put forward. (shrink)

The broadly-stated aim of this rich collection is to reevaluate and reconceptualize the mathematization thesis, which the editors take to signify “above all the transformation of scientific concepts and methods, especially those concerning the nature of matter, space, and time, through the introduction of mathematical techniques and ideas”. As a historiographical thesis, it is the thesis that “the scientific revolution, and by implication modern science as a whole, is guided by the project of mathematization”.In the introduction to the volume, the (...) editors acknowledge the virtues of the historiographical thesis, which explain its persistence. For example, it highlights a constitutive feature... (shrink)

Language, through the discrete nature of linguistic names and strictly determined grammatical rules, creates absolute, “quantized”, sharply separated “facts” within the external world that is continuous, “fuzzy” and relational in its essence. Therefore, it is similar, in some important sense, to magic, which attributes causal and creative power to magical words and formulas. On the one hand, language increases greatly the effectiveness of the processes of thinking and interpersonal communication, yet, on the other hand, it determines and distorts (...) to a large extent the picture of the world created within the mind. The relatively smallest (but still significant) magical admixture is present in science, because of its relatively reliable methodology, while the largest is found in religion and a large part of philosophy. The magical nature of language also manifests itself in logic and mathematics that refer to ill determined, fuzzy objects, sets and relations in the real world. The meaning of linguistic names is based on the conceptual network—an epiphenomenon (continuous in its essence) of the neural network—where interactions between particular concepts are based on the relation of connotation. The names and formulas of language correspond to these concepts which are best separated and determined. A direct relation of denotation between the elements of language and “facts” of the world is an illusion. While we cannot dispense with language because of its immense usefulness, we must remember about its “fact-creating” nature and influence on our thought and cognitive processes. The picture of the reality created as the result of them is to a large extent formed and deformed by the nature of language, and not by the “immanent” properties of the world in itself. (shrink)

Although the mathematization of nature is a distinctive and crucial feature of the emergence of modern science in the seventeenth century, this volume shows that it was a far more complex, contested, and context-dependent phenomenon than the received historiography has indicated.0.

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Michael Dummett is a leading contemporary philosopher whose work on the logic and metaphysics of language has had a lasting influence on how these subjects are conceived and discussed. This volume contains some of the most provocative and widely discussed essays published in the last fifteen years, together with a number of unpublished or inaccessible writings. Essays included are: "What is a Theory of Meaning?," "What do I Know When I Know a Language?," "What Does the Appeal to (...) Use Do for the Theory of Meaning?," "Language and Truth," "Truth and Meaning," "Language and Communication," "The Source of the Concept of Truth," "Mood, Force, and Convention," "Frege and Husserl on Reference," "Realism," "Existence," "Does Quantification Involve Identity?," "Could there be Unicorns?," "Causal Loops," "Common Sense and Physics," "Testimony and Memory," "What is Mathematics About?," "Wittgenstein on Necessity: Some Reflections," and "Realism and Anti-Realism." Serving well as a companion to Dummett's other collections, the essays in this volume are not forbiddingly technical or specialized, and have relevance to many areas of analytic philosophy. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

A common description of a mathematical proof is as a logically structured sequence of assertions, beginning from accepted premises and proceeding by standard inference rules to a conclusion. Does this description match the language of proofs as mathematicians write them in their research articles? In this chapter, we use methods from corpus linguistics to look at the prevalence of imperatives and instructions in mathematical preprints from the arXiv repository. We find thirteen verbs that are used most often to form (...) imperatives in proofs, and that these show up significantly more often within proofs than in the surrounding mathematical writing. We also show that there are many more verbs used to form a diverse selection of instructions in proofs. These findings are at odds with the view of proofs as sequences of assertions. Instead, we argue in favour of the recipe model of proofs: that proofs are like recipes, giving instructions for mathematical actions to be carried out. (shrink)

What do ‘meaning’ and ‘truth’ mean? And how are they situated in the concrete practices of linguistic communication? What is the relationship between words and the world? How—with words—can people do such varied things as marry, inaugurate a president, and declare a country’s independence? How is language able to express knowledge, belief, and other mental states? What are metaphors and how do they work? Is a mathematically rigorous account of language possible? Does language make women invisible and (...) encode a male worldview? These are the kind of questions that have been addressed by philosophers of language since ancient times. Interest in the subject stretches back to the beginnings of western philosophy . Interest in the philosophy of language has also been enduring—and has blossomed anew in the past century. This new title in the Routledge series, Critical Concepts in Philosophy, meets the need for an authoritative reference work to make sense of the subject’s vast literature and the continuing explosion in research output. Edited by A. P. Martinich, one of the subdiscipline’s leading scholars, this collection brings together in four volumes the canonical and the very best cutting-edge scholarship in the field to provide a synoptic view of all the key issues, figures, concepts, and current debates. With comprehensive introductions to each volume, newly written by the editor, which place the collected material in its historical and intellectual context, Philosophy of Language is an essential work of reference and is destined to be valued by philosophers and linguists—as well as psychologists and cognitive scientists working on language—as a vital research resource. (shrink)

In this paper, we propose two agent-based simulation models for the evolution of language in the framework of evolutionary language games. The theory of evolutionary language games arose from the union of evolutionary game theory, introduced by the English biologist John Maynard Smith, and language games, developed by the Austrian philosopher Ludwig Wittgenstein. The first model proposed is based on Martin Nowak’s work and is designed to reproduce and verify the results Nowak obtained in his simplest (...) mathematical model. For the second model, we extend the previous one with the introduction of a world where the languages live and evolve, and which influences interactions among individuals. The main goal of this research is to present a model which shows how the presence of a topological structure influences the communication among individuals and contributes to the emergence of clusters of different languages. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. (...) Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

Despite progress, gender gaps persist in mathematical and language-related fields, and gender stereotypes likely play a role. The current study examines the relations between parents’ gender-related beliefs and their adolescent child’s motivation and career aspirations through a survey of 172 parent-child dyads. Parents reported their gendered beliefs about ability in mathematics and language arts, as well as their prescriptive gender role beliefs. Students reported their expectancies and values in these two domains, as well as their career aspirations (...) The results of path models suggested that parents’ ability stereotypes about language boosted girls’ motivation for language arts, thereby nudging them away from STEM pathways. Girls’ career aspirations stemmed not only from their valuation of the corresponding domain, but also from their valuation of competing domains. Such findings highlight the need to consider multiple domains simultaneously in order to better capture the complexity of girls’ career decisions. For boys, parents’ language ability stereotypes were directly related to mathematical career aspirations. These results suggest that stereotypes that language arts is not for boys push them instead toward mathematics. Our study also highlighted the unique role of parental beliefs in traditional gender roles for boys’ motivation and career aspirations. Specifically, parents’ gender role stereotypes directly related to less interest in language arts only among boys. This highlights that research into gender gaps in female-dominated fields should consider stereotypes related to appropriate behavior and social roles for boys. (shrink)

The aim of this article is to present and discuss the language of the «givens», a typical stylistic resource of Greek mathematics and one of the major features of the proof format of analysis and synthesis. I shall analyze its expressive function and its peculiarities, as well as its general role as a deductive tool, explaining at the same time its particular applications in subgenres of a geometrical proposition like the locus theorems and the so-called «porisms». The main (...) interpretative theses of this study are the following: the language of the «givens» (1) is the standard idiom in which “existence and uniqueness” of a mathematical object was proved, (2) was conceived as an unified framework reducing to a strictly deductive format disparate argumentative steps such as deductions, constructions, and calculations. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in (...) this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

The chapters in this timely volume aim to answer the growing interest in Arthur Schopenhauer’s logic, mathematics, and philosophy of language by comprehensively exploring his work on mathematical evidence, logic diagrams, and problems of semantics. Thus, this work addresses the lack of research on these subjects in the context of Schopenhauer’s oeuvre by exposing their links to modern research areas, such as the “proof without words” movement, analytic philosophy and diagrammatic reasoning, demonstrating its continued relevance to current discourse (...) on logic. -/- Beginning with Schopenhauer’s philosophy of language, the chapters examine the individual aspects of his semantics, semiotics, translation theory, language criticism, and communication theory. Additionally, Schopenhauer’s anticipation of modern contextualism is analyzed. The second section then addresses his logic, examining proof theory, metalogic, system of natural deduction, conversion theory, logical geometry, and the history of logic. Special focus is given to the role of the Euler diagrams used frequently in his lectures and their significance to broader context of his logic. In the final section, chapters discuss Schopenhauer’s philosophy of mathematics while synthesizing all topics from the previous sections, emphasizing the relationship between intuition and concept. -/- Aimed at a variety of academics, including researchers of Schopenhauer, philosophers, historians, logicians, mathematicians, and linguists, this title serves as a unique and vital resource for those interested in expanding their knowledge of Schopenhauer’s work as it relates to modern mathematical and logical study. (shrink)

Language and information processes are critical issues in scientific controversies regarding the qualities that epitomize humanness. Whereas some theorists claim human mental uniqueness with regard to language, others point to successes in teaching language skills to other animals. However, although these animals may learn names for things, they show little ability to utilize a complex framework of symbolic reference. In such a framework, words or other symbols refer not only to objects and concepts but also to sequential (...) and hierarchical relationships with other symbols. This process is essential to human mental operations, including language, mathematics, and music. In humans, these operations may have coevolved with the prefrontal area of the cerebral cortex, which is proportionately much larger in humans than in other animals and more intricately linked with other areas of the brain. Analysis of the structure and function of the prefrontal area suggests that it is centrally involved in the operation of higher‐order associative relationships involving the subordination of one set of associations to another. This alternate learning strategy apparently appeared at the cost of certain sensory, motor, or limbic abilities. The payoff was symbolic thinking. Humans thus are unique among species, not just for their highly developed language ability but for their odd style of thinking and learning. (shrink)

Excerpt from The Mathematical Psychology of Gratry and Boole: Translated From the Language of the Higher Calculus Into That of Elementary Geometry Dear Dr. Maudsley, - You have often asked me to explain, for students unaquainted with the Infinitesimal Calculus, certain doctrines expressed in terms of that Calculus by P. Gratry and my late husband. That you permit me to dedicate my attempt to you will, at least, be a guarantee that the main ideas of mathematical psychology are based, (...) not on mystic dreams, but on scientific induction. I am glad of this opportunity of expressing to you my gratitude, both for your published works, and for much personal kindness to myself. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

The paper proposes a reflection on mathematical beauty, considering the possibility of aesthetic qualities for formal language. Through a concise overview of the way this question is understood by some famous scientists and mathematicians, we turn our attention to Gian-Carlo Rota’s theoretical proposal: his reflections as a mathematician and philosopher offer a perspective, of phenomenological matrix, fruitful for looking at the question. Rota’s contribution allows us to focus on the role of competence, acquired through effort, sedimentation and habit of (...) repetition, in cultivating the potential to recognise the aesthetic quality of formal language. Finally, we draw on some contributions from Gestalt theory, closely connected to twentieth-century phenomenology for chronological and conceptual reasons, applying the idea of gestalt wholeness to the mathematical product and proposing some reflections that may help to clarify some of the dynamics underlying the attribution of qualities of beauty to formulas or the detection of its lack. (shrink)

Since the first chapter of this book presents an intro duction to the present state of game-theoretical semantics (GTS), there is no point in giving a briefer survey here. Instead, it may be helpful to indicate what this volume attempts to do. The first chapter gives a short intro duction to GTS and a survey of what is has accomplished. Chapter 2 puts the enterprise of GTS into new philo sophical perspective by relating its basic ideas to Kant's phi losophy (...) of mathematics, space, and time. Chapters 3-6 are samples of GTS's accomplishments in understanding different kinds of semantical phenomena, mostly in natural languages. Beyond presenting results, some of these chapters also have other aims. Chapter 3 relates GTS to an interesting line of logical and foundational studies - the so-called functional interpretations - while chapter 4 leads to certain important methodological theses. Chapter 7 marks an application of GTS in a more philo sophical direction by criticizing the Frege-Russell thesis that words like "is" are multiply ambiguous. This leads in turn to a criticism of recent logical languages (logical notation), which since Frege have been based on the ambi guity thesis, and also to certain methodological sug gestions. In chapter 8, GTS is shown to have important implications for our understanding of Aristotle's doctrine of categories, while chapter 9 continues my earlier criticism of Chomsky's generative approach to linguistic theorizing. (shrink)

I give an informal presentation of the evolutionary game theoretic approach to the conventions that constitute linguistic meaning. The aim is to give a philosophical interpretation of the project, which accounts for the role of game theoretic mathematics in explaining linguistic phenomena. I articulate the main virtue of this sort of account, which is its psychological economy, and I point to the casual mechanisms that are the ground of the application of evolutionary game theory to linguistic phenomena. Lastly, I (...) consider the objection that the account cannot explain predication, logic, and compositionality. (shrink)

In this paper, I survey and defend the use of quantitative methods in philosophy of language. Quantitative methods in philosophy of language include a wide variety of methods, ranging from model‐based techniques (computer simulations and mathematical models) to data‐driven approaches (experimental philosophy and corpus‐based studies). After offering a few case studies of these methodologies in action, I single out some debates in philosophy of language that are especially well served by their use. These are cases in which (...) quantitative methods increase precision, improve the accuracy and reliability of results, and allow philosophers of language to borrow from other fields the best epistemic practices available. I conclude with some far‐reaching considerations about the role of qualitative and quantitative methods in debates about philosophical methodology. (shrink)

Logicism and the Philosophy of Language brings together the core works by Gottlob Frege and Bertrand Russell on logic and language. In their separate efforts to clarify mathematics through the use of logic in the late nineteenth and early twentieth century, Frege and Russell both recognized the need for rigorous and systematic semantic analysis of language. It was their turn to this style of analysis that would establish the philosophy of language as an autonomous area (...) of inquiry. This anthology gathers together these foundational writings, and frames them with an extensive historical introduction. This is a collection for anyone interested in questions about truth, meaning, reference, and logic, and in the application of formal analysis to these concepts. (shrink)

How does mathematics apply to something non-mathematical? We distinguish between a general application problem and a special application problem. A critical examination of the answer that structural mapping accounts offer to the former problem leads us to identify a lacuna in these accounts: they have to presuppose that target systems are structured and yet leave this presupposition unexplained. We propose to fill this gap with an account that attributes structures to targets through structure generating descriptions. These descriptions are physical (...) descriptions and so there is no such thing as a solely mathematical account of a target system. (shrink)