Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language (...) semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. (shrink)

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in (...) terms of scientific modelling. I argue against both the Conceptualist and Platonist interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics. (shrink)

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less (...) prevalent in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

Wales uses languages with both regular (Welsh) and irregular (English) counting systems. Three groups of 6- and 8-year-old Welsh children with varying degrees of exposure to the Welsh language—those who spoke Welsh at both home and school; those who spoke Welsh only at home; and those who spoke only English—were given standardized tests of arithmetic and a test of understanding representations of two-digit numbers. Groups did not differ on the arithmetic tests, but both groups of Welsh speakers read and compared (...) 2-digit numbers more accurately than monolingual English children. A similar study was carried out with Tamil/English bilingual children in England. The Tamil counting system is more transparent than English but less so than Welsh or Chinese. Tamil-speaking children performed better than monolingual English-speaking children on one of the standardized arithmetic tests but did not differ in their comparison of two-digit numbers. Reasons for the findings are discussed. (shrink)

La théorie leibnizienne de l'expression, centrée sur la notion de relation, introduit, entre les mots des langues naturelles et la pensée, un rapport qui n'est pas seulement de représentation. Elle introduit également une parenté entre langues naturelles et langages formels. L'objectif de l'article est de mener une confrontation entre l'analyse par Leibniz des relations dans les langues naturelles et dans les langages symboliques afin de mettre en évidence leurs analogies. L'article cherchera à montrer : l'existence d'une double articulation dans les (...) langues naturelles et les langages formels tels que les conçoit Leibniz ; le caractère mathématique de son analyse des prépositions ; l'arrière-plan logique de son analyse des dérivations sémantiques. (shrink)

This book is dedicated to the life and work of the mathematician Joachim Lambek. The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist. In the 1960s, he began to work in category theory, categorical algebra, logic, (...) proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development, he introduced a novel and deeper algebraic framework for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists. (shrink)

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the (...) linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

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)

The formal sciences, particularly mathematics, have had a profound influence on the development of linguistics. This insightful overview looks at techniques that were introduced in the fields of mathematics, logic and philosophy during the twentieth century, and explores their effect on the work of various linguists. In particular, it discusses the 'foundations crisis' that destabilised mathematics at the start of the twentieth century, the numerous related movements which sought to respond to this crisis, and how they influenced the development (...) of syntactic theory in the 1950s. The book concludes by discussing the resulting major consequences for syntactic theory, and provides a detailed reassessment of Chomsky's early work at the advent of Generative Grammar. Informative and revealing, this book will be invaluable to all those working in formal linguistics, in particular those interested in its history and development. (shrink)

"Foundations of the Formal Sciences" is a series of interdisciplinary conferences in mathematics, philosophy, computer science and linguistics. The main goal is to reestablish the traditionally strong links between these areas of research that have been lost in the past decades. The second conference in the series had the subtitle "Applications of Mathematical Logic in Philosophy and Linguistics" and brought speakers from all parts of the Formal Sciences together to give a holistic view of how mathematical (...) methods can improve our philosophical and technical understanding of language and scientific discourse, ranging from the theoretical level up to applications in language recognition software. Audience: This volume is of interest to all formal philosophers and theoretical linguists. In addition to that, logicians interested in the applications of their field and logic students in mathematics, computer science, philosophy and linguistics can use the volume to broaden their knowledge of applications of logic. (shrink)

In previous work we gave a mathematical foundation, referred to as DisCoCat, for how words interact in a sentence in order to produce the meaning of that sentence. To do so, we exploited the perfect structural match of grammar and categories of meaning spaces. Here, we give a mathematical foundation, referred to as DisCoCirc, for how sentences interact in texts in order to produce the meaning of that text. First we revisit DisCoCat. While in DisCoCat all meanings are (...) fixed as states, in DisCoCirc word meanings correspond to a type, or system, and the states of this system can evolve. Sentences are gates within a circuit which update the variable meanings of those words. Like in DisCoCat, word meanings can live in a variety of spaces e.g. propositional, vectorial, or cognitive. The compositional structure are string diagrams representing information flows, and an entire text yields a single string diagram in which word meanings lift to the meaning of the entire text. While the developments in this paper are independent of a physical embodiment, both the compositional formalism and suggested meaning model are highly quantum-inspired, and implementation on a quantum computer would come with a range of benefits. We also praise Jim Lambek for his role in mathematical linguistics in general, and the development of the DisCo program more specifically. (shrink)

Classes of linguistic paradoxes are introduced with examples and explanations. They are part of the author's work on the Paradoxist Philosophy based on mathematical logic.

This is the first book-length analysis of the techniques and procedures of ancient mathematical commentaries. It focuses on examples in Chinese, Sanskrit, Akkadian and Sumerian, and Ancient Greek, presenting the general issues by constant detailed reference to these commentaries, of which substantial extracts are included in the original languages and in translation, sometimes for the first time. This makes the issues accessible to readers without specialized training in mathematics or in the languages involved. The result is a much richer (...) understanding than was hitherto possible of the crucial role of commentaries in the history of mathematics in four different linguistic areas, of the nature of mathematical commentaries in general, of the contribution that the study of mathematical commentaries can make to the history of science and to the study of commentaries in general, and of the ways in which mathematical commentaries are like and unlike other kinds of commentaries. (shrink)

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae in Greek mathematical texts. It is shown that the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that (...) such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level is the best explanatory level for the phenomenon of Greek mathematical deduction. (shrink)

In this paper the authors develop a logic of concepts within a mathematical linguistic theory. In the set of concepts defined in a belief system, the order relationship and Boolean algebra of the concepts are considered. This study is designed to obtain a tool, which is the metatheoretical base of this type of theory.

Reality contains information that becomes significances in the mind of the observer. Language is the human instrument to understand reality. But is it possible to attain this reality? Is there an absolute reality, as certain philosophical schools tell us? The reality that we perceive, is it just a fragmented reality of which we are part? The work that the authors present is an attempt to address this question from an epistemological, linguistic and logical-mathematical point of view.

Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary (...) philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 9.2 (2002) 127-129 [Access article in PDF] Linguistic Markers of Recovery:Underpinnings of First Person Pronoun Usage and Semantic Positions of Patients Patrick Suppes Keywords: association, freedom, habits, psychotherapy, roles, semantics. USING LINGUISTIC EVIDENCE to evaluate recovering psychotherapy patients is an attractive and useful idea. I agree with much of Dr. van Staden's proposals for doing so. The purpose of this commentary is to give some (...) comments and criticisms on points of detail. Roles, Not Semantics A fair amount of Dr. van Staden's article is concerned with the semantics of relations. The focus on relations is a good thing. Restricting discourse about patients to properties is not sufficient, for everyone regards how a person is relating to others, and even to other things, as an important feature of normal behavior and mental health. We do not want just to report, "John was upset," but consider it definitely more informative to say, "John was upset by his father's criticisms." The English language and all other natural languages are rich in verbs for expressing relations with emotional affect between two or more persons, or a person and an event or thing. Staden's article focuses on the interesting and important case of first person pronouns. To go from saying, "She made me do it" to "I did it" is, for reasons that need not be spelled out, a sign of progress. A way to describe this change is that the speaker has indicated that in the implicit context assumed he or she has shown signs of moving from the role of being a patient to that of being an actor, from being a recipient of actions to being a performer of acts.van Staden concentrates on the semantics of relations to bring out what I call the change of roles. I would certainly agree that such change of roles is important to the semantics of affective or action verbs and verb phrases. We have no disagreement at all about this. Yet I am critical of his wanting to range much wider in his analysis of semantics. The logical or formal treatment of relations is a large topic, full of many distinctions and subtleties that seem to me not too relevant to his main theme. In a very general way he wants to say that what is usually thought of as [End Page 127] the first position in a relation, what he calls the alpha position, is occupied by the owner of the relation, and the second position, in a binary relation—the main case discussed—is occupied by the accidental to that particular relation. This second position is called the omega position.This runs contrary to much of the mathematical literature on relations, which occupy an important place in many parts of mathematics such as geometry. So when we say that one triangle is larger than another, it seems positively weird to say the second triangle is somehow accidental to the relation. But this is true and confusing even for the familiar and classical kinds of actions often cited by van Staden. In analyzing the sentence "Brutus killed Caesar intentionally" it sounds really strange to say that Caesar occupied the accidental position in the expressed relation of killing.Moreover, in the conceptual and philosophical discussion of relations, it is common practice to stress that even if the natural grammar of a sentence has a subject-predicate structure, as in "Elizabeth is the mother of Charles," no special logical force is given to what van Staden calls the first or alpha position in most mathematical relations, such as "The number 2 is less than the number 3," expressed in ordinary mathematical notation as "2 < 3."On the other hand, I stress that it is natural to distinguish in a relational sentence expressing an action between the role of actor and the role of patient, no matter which is the grammatical subject of the sentence, as in the following pair "John hurt Mary" and "Mary was hurt by John." The roles of actor... (shrink)

The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our attention to (...) experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods. (shrink)

Linguistics as a science has rapidly changed during the course of a relatively short period. The mathematical foundations of the science, however, present a different story below the surface. In this paper, I argue that due to the former, the seismic shifts in theory over the past 80 years opens linguistics up to the problem of pessimistic meta-induction or radical theory change. I further argue that, due to the latter, one current solution to this problem in the (...) philosophy of science, namely structural realism :403–424, 1998; French in Proc Aristot Soc 106:167–185, 2006), should be viewed as especially enticing for linguists, as their field is a largely structural enterprise. I discuss particular historical instances of theory change in generative syntax before investigating two views on the nature of structural properties and eventually proposing an approach in terms of invariance :162–186, 2015) as a grounding for structural realism in the history and philosophy of linguistics. (shrink)

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

What is a ‘number,’ as studied within numerical cognition? The term is highly polysemous, and can refer to numerals, numerosity, and a diverse collection of mathematical objects, from natural numbers to infinitesimals. However, numerical cognition has focused primarily on prototypical counting numbers – numbers used regularly to count small collections of objects. Even these simple numbers are far more complex than apparent pre-conditions for numerical abilities like subitizing and approximate discrimination of large numerosity, which we share with other animals. (...) We argue that the leap to number concepts proper relies, in part, on two embodied, domain-general cognitive mechanisms: conceptual metaphor and fictive motion. These mechanisms were first investigated within cognitive linguistics, a subdiscipline of cognitive science, but are now thought to subserve cognition more generally. We review the proposal that these mechanisms structure numerical cognition – including PCNs, but also the positive integers and arithmetic – and survey the supporting empirical evidence. (shrink)

Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill corpus, or epistemic aims of (...) mathematicians or mathematical communities. We focus on the relation between explanation, epistemic aims and diverging explanatory assessments by looking at cases from mathematical practice. (shrink)

Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.

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 paper, we introduce a suppositional view of linguistic practice that ranges over fiction, science, and mathematics. While having similar con- sequences to some other views, in particular Linsky and Zalta’s plenitudinous platonism, the view advocated here both differs fundamentally in approach and accounts for a wider range of phenomena and scientific discourse.

In this book, for the first time, authors try to introduce the concept of linguistic variables as a continuum of linguistic terms/elements/words in par or similar to a real continuum. For instance, we have the linguistic variable, say the heights of people, then we place the heights in the linguistic continuum [shortest, tallest] unlike the real continuum (–∞, ∞) where both –∞ or +∞ is only a non-included symbols of the real continuum, but in case of the linguistic continuum we (...) generally include the ends or to be more mathematical say it is a closed interval, where shortest denotes the shortest height of a person, maybe the born infant who is very short from usual and the tallest will denote the tallest one usually very tall; however this linguistic continuum [shortest, tallest] in the real continuum will be the closed interval say [1 foot, 8 feet] or [1, 8] the measurement in terms of feet. So, the real interval is a subinterval with which we have associated the real continuum in terms of qualifying unit feet and inches in this case. (shrink)

Chapter Some topics in semantics Aims of this study The central preoccupation of this study is semantic. It is intended as a modest contribution to the ...

Advances in modern mathematics indicate that progress in this field of knowledge depends mainly on culturally inflected imaginative intuitions, or intuitive imaginings—which mysteriously result in the growth of systems of symbolism that are often efficacious, although fallible and very likely evolutionary. Thus the idea that a trouble-free epistemology can be constructed out of an intuition-free mathematical naturalism would seem to be question begging of a very high order. I illustrate the point by examining Philip Kitcher’s attempt to frame an (...) empiricist philosophy of mathematics, which he calls “mathematical naturalism,” wherein he proposes to explain novelty in mathematics by means of the notion of ‘rational interpractice transitions,’ only to end with an appeal to science to supply a meaning for rationality. A more promising naturalistic approach is adumbrated by Noam Chomsky who begins with a straightforward acceptance of mind and language as ‘natural’ or concrete facts which bespeak the need for a linguistic faculty. This indicates in turn that there may also be a mathematical faculty capable of generating and exploiting the powers of mathematical symbolisms in a manner analogous to the linguistic faculty. (shrink)

This paper argues that the shared intersubjective accessibility of mathematical objects has its roots in a stratum of experience prior to language or any other form of concrete social interaction. On the basis of Husserl’s phenomenology, I demonstrate that intersubjectivity is an essential stratum of the objects of mathematical experience, i.e., an integral part of the peculiar sense of a mathematical object is its common accessibility to any consciousness whatsoever. For Husserl, any experience of an objective nature (...) has as its correlate a “we,” which he terms the “community of monads”. Thus, even before mathematical objects gain expression, formalization, and axiomatization through natural and scientific language, from a phenomenological viewpoint their objectivity has its roots in raw pre-linguistic though intersubjective experience. Accordingly, I demonstrate the different senses in which the experience of mathematical objects is permeated by intersubjectivity, suggesting a picture of mathematical intersubjectivity as pre-linguistic common experience based on Husserl’s idea of a “community of monads”. (shrink)