The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.
Mind–body dualism has rarely been an issue in the generative study of mind; Chomsky himself has long claimed it to be incoherent and unformulable. We first present and defend this negative argument but then suggest that the generative enterprise may license a rather novel and internalist view of the mind and its place in nature, different from all of, (i) the commonly assumed functionalist metaphysics of generative linguistics, (ii) physicalism, and (iii) Chomsky’s negative stance. Our argument departs from the (...) empirical observation that the linguistic mind gives rise to hierarchies of semantic complexity that we argue (only) follow from constraints of an essentially mathematical kind. We assume that the faculty of language tightly correlates with the mathematical capacity both formally and in evolution, the latter plausibly arising as an abstraction from the former, as a kind of specialized output. On this basis, and since the semantic hierarchies in question are mirrored in the syntactic complexity of the expression involved, we posit the existence of a higher-dimensional syntax structured on the model of the hierarchy of numbers, in order to explain the semantic facts in question. If so, syntax does not have a physicalist interpretation any more than the hierarchy of number-theoretic spaces does. (shrink)
Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in (...) that it is much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (shrink)
Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a (...) derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optinal sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science. (shrink)
A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...) Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)
Chomsky’s highly influential Syntactic Structures ( SS ) has been much praised its originality, explicitness, and relevance for subsequent cognitive science. Such claims are greatly overstated. SS contains no proof that English is beyond the power of finite state description (it is not clear that Chomsky ever gave a sound mathematical argument for that claim). The approach advocated by SS springs directly out of the work of the mathematical logician Emil Post on formalizing proof, but few linguists are (...) aware of this, because Post’s papers are not cited. Chomsky’s extensions to Post’s systems are not clearly defined, and the arguments for their necessity are weak. Linguists have also overlooked Post’s proofs of the first two theorems about effects of rule format restrictions on generative capacity, published more than ten years before SS was published. (shrink)
Jim Brown (1991, viii) says that platonism, in mathematics involves the following: 1. mathematical objects exist independently of us; 2. mathematical objects are abstract; 3. we learn about mathematical objects by the faculty of intuition. The same is being claimed by Jerrold Katz (1981, 1998) in his platonistic approach to linguistics. We can take the object of linguistic analysis to be concrete physical sounds as held by nominalists, or we can assume that the object of linguistic (...) study are psychological or mental states which presents the conceptualism or psychologism of Chomsky and that language is an abstract object as held by platonists or realists and urged by Jerrold Katz hinlself.I want to explicate Katz’s proposal which is based on Kant’s conception of pure intuition and give arguments why I find it implausible. I also present doubts that linguists use intuitive evidence only. I conclude with some arguments against the a prioricity of intuitive judgements in general which is also relevant for Jim Brown’s platonistic beliefs. (shrink)
In this paper, I discuss the social philosopher Pierre Bourdieu’s concept of habitus, and use it to locate and examine dispositions in a larger constellation of related concepts, exploring their dynamic relationship within the social context, and their construction, manifestation, and function in relation to classroom mathematics practices. I describe the main characteristics of habitus that account for its invisible effects: its embodiment, its deep and pre-reflective internalization as schemata, orientation, and taste that are learned and yet unthought, and are (...) subsumed by our practices, which we understand as something that “goes without saying.” I also propose that, similarly to Bourdieu’s concept of linguistic habitus, a math habitus is made up of a complex intertwining of collective and individual histories that turn into “nature,” which structure all individual and collective action and inform mathematical classroom practice. I suggest that individual math dispositions may be liable to reconstruction through the reconstruction of the collective math habitus, which follows from opening spaces for dialogue, problematization and reconstruction of the unthought categories of the doxa. This requires that students acquire new concrete and symbolic means with which to challenge their current sense of mathematics as a discipline, and mathematical practice tout court. Finally, I argue that community of inquiry, employed as a pedagogical model, provides an avenue for both: for opening those spaces for reflective dialogical inquiry into concepts and questions whose meanings and references have so far been taken for granted, and for acquiring critical thinking and dialogical skills and dispositions that are a necessary means for participating in such reflective inquiry that offers significant promise for reconstructing individual and collective habitus in school settings. (shrink)
Open peer commentary on the target article “How and Why the Brain Lays the Foundations for a Conscious Self” by Martin V. Butz. Excerpt: In this commentary to Martin V. Butz’s target article I am especially concerned with his remarks about language (§33, §§71–79, §91) and modularity (§32, §41, §48, §81, §§94–98). In that context, I would like to bring into discussion my own work on computational models of self-monitoring (cf. Neumann 1998, 2004). In this work I explore the idea (...) of an anticipatory drive as a substantial control device for modelling high-level complex language processes such as selfmonitoring and adaptive language use. My work is grounded in computational linguistics and, as such, uses a mathematical and computational methodology. Nevertheless, it might provide some interesting aspects and perspectives for constructivism in general, and the model proposed in Butz’s article, in particular. (shrink)
The dominant approach to analyzing the meaning of natural language sentences that express mathematical knowl- edge relies on a referential, formal semantics. Below, I discuss an argument against this approach and in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits, including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma. The new perspective also promises to facilitate discussion between philosophers of mathematics and (...) cognitive scientists working on topics of common interest. (shrink)
Linguists take the intuitive judgments of speakers to be good evidence for a grammar. Why? The Chomskian answer is that they are derived by a rational process from a representation of linguistic rules in the language faculty. The paper takes a different view. It argues for a naturalistic and non-Cartesian view of intuitions in general. They are empirical central-processor responses to phenomena differing from other such responses only in being immediate and fairly unreflective. Applying this to linguistic intuitions yields an (...) explanation of their evidential role without any appeal to the representation of rules. Introduction The evidence for linguistic theories A tension in the linguists' view of intuitions Intuitions in general Linguistic intuitions Comparison of the modest explanation with the standard Cartesian explanation A nonstandard Cartesian explanation of the role of intuitions? Must linguistics explain intuitions? Conclusion. (shrink)
Michael Devitt has argued that Chomsky, along with many other Linguists and philosophers, is ignorant of the true nature of Generative Linguistics. In particular, Devitt argues that Chomsky and others wrongly believe the proper object of linguistic inquiry to be speakers' competences, rather than the languages that speakers are competent with. In return, some commentators on Devitt's work have returned the accusation, arguing that it is Devitt who is ignorant about Linguistics. In this note, I consider whether there (...) might be less to this apparent dispute than meets the eye. -/- . (shrink)
This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) class='Hi'>mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)
Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...) are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...) with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)
Philosophy of linguistics is the philosophy of science as applied to linguistics. This differentiates it sharply from the philosophy of language, traditionally concerned with matters of meaning and reference.
The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness (...) and completeness -- Why is first order logic "First Order"? (shrink)
This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a (...) human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection” to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. (shrink)
Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.
In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to several morals (...) about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)
Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or (...) creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)
The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...) with successful representation, scientists often rely on the existence of a ‘mature mathematical formalism’, where the latter refers to a—mathematically formulated and physically interpreted—notational system of locally applicable rules that derive from (but need not be reducible to) fundamental theory. As mathematical formalisms undergo a process of elaboration, enrichment, and entrenchment, they come to embody theoretical, ontological, and methodological commitments and assumptions. Since these are enshrined in the formalism itself, they are no longer readily obvious to either the novice or the proficient user. At the same time as formalisms constrain what may be represented, they also function as inferential and interpretative resources. (shrink)
Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give (...) a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs. (shrink)
An influential position in the philosophy of biology claims that there are no biological laws, since any apparently biological generalization is either too accidental, fact-like or contingent to be named a law, or is simply reducible to physical laws that regulate electrical and chemical interactions taking place between merely physical systems. In the following I will stress a neglected aspect of the debate that emerges directly from the growing importance of mathematical models of biological phenomena. My main aim is (...) to defend, as well as reinforce, the view that there are indeed laws also in biology, and that their difference in stability, contingency or resilience with respect to physical laws is one of degrees, and not of kind. In order to reach this goal, in the next sections I will advance the following two arguments in favor of the existence of biological laws, both of which are meant to stress the similarity between physical and biological laws. (shrink)
It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical know- ledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition.’ On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations results in true beliefs. This view is free (...) of any conditions incompatible with abstract objects, for the reason that it is not necessary that S stand in some causal relation to the entities in virtue of which p is true. Mathematical intuition is simply one kind of reliable process type, whose inputs are not abstract numbers, but rather, contemplations of abstract numbers. (shrink)
To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement (...) about ‘truth’, but the assignment of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)
Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...) II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)
This lively introduction to mathematical logic, easily accessible to non-mathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursivefunctions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.
Robert Stalnaker has argued that mathematical information is information about the sentences and expressions of mathematics. I argue that this metalinguistic account is open to a variant of Alonzo Church's translation objection and that Stalnaker's attempt to get around this objection is not successful. If correct, this tells not only against Stalnaker's account of mathematical truths, but against any metalinguistic account of truths that are both necessary and informative.
Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
This article suggests that scientific philosophy, especially mathematical philosophy, might be one important way of doing philosophy in the future. Along the way, the article distinguishes between different types of scientific philosophy; it mentions some of the scientific methods that can serve philosophers; it aims to undermine some worries about mathematical philosophy; and it tries to make clear why in certain cases the application of mathematical methods is necessary for philosophical progress.
One of the central issues in linguistics is whether or not language should be considered a self-contained, autonomous formal system, essentially reducible to the syntactic algorithms of meaning construction (as Chomskyan grammar would have it), or a holistic-functional system serving the means of expressing pre-organized intentional contents and thus accessible with respect to features and structures pertaining to other cognitive subsystems or to human experience as such (as Cognitive Linguistics would have it). The latter claim depends critically on (...) the existence of principles governing the composition of semantic contents. Husserl''s fourth Logical Investigation is well known as a genuine precursor for Chomskyan grammar. However, I will establish the heterogeneous character of the Investigation and show that the whole first part of it is devoted to the exposition of a semantic combinatorial system cognate to the one elaborated within Cognitive Linguistics. I will thus show how theoretical results in linguistics may serve to corroborate and shed light on those parts of Husserl''s Fourth Investigation that have traditionally been dismissed as vague or simply ignored. (shrink)
This book deals with the need to rethink the aims and methods of contemporary linguistics. Orthodox linguists' discussions of linguistic form fail to exemplify how language users become language makers. Integrationist theory is used here as a solution to this basic problem within general linguistics. The book is aimed at an interdisciplinary readership, comprising those engaged in study, teaching and research in the humanities and social sciences, including linguistics, philosophy, sociology and psychology.