This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information and inspiration for (...) future research on this topic. (shrink)

We model a piece of text of human language telling a story by means of the quantum structure describing a Bose gas in a state close to a Bose–Einstein condensate near absolute zero temperature. For this we introduce energy levels for the words (concepts) used in the story and we also introduce the new notion of ‘cogniton’ as the quantum of human thought. Words (concepts) are then cognitons in different energy states as it is the case for photons in different (...) energy states, or states of different radiative frequency, when the considered boson gas is that of the quanta of the electromagnetic field. We show that Bose–Einstein statistics delivers a very good model for these pieces of texts telling stories, both for short stories and for long stories of the size of novels. We analyze an unexpected connection with Zipf’s law in human language, the Zipf ranking relating to the energy levels of the words, and the Bose–Einstein graph coinciding with the Zipf graph. We investigate the issue of ‘identity and indistinguishability’ from this new perspective and conjecture that the way one can easily understand how two of ‘the same concepts’ are ‘absolutely identical and indistinguishable’ in human language is also the way in which quantum particles are absolutely identical and indistinguishable in physical reality, providing in this way new evidence for our conceptuality interpretation of quantum theory. (shrink)

We model a piece of text of human language telling a story by means of the quantum structure describing a Bose gas in a state close to a Bose–Einstein condensate near absolute zero temperature. For this we introduce energy levels for the words used in the story and we also introduce the new notion of ‘cogniton’ as the quantum of human thought. Words are then cognitons in different energy states as it is the case for photons in different energy states, (...) or states of different radiative frequency, when the considered boson gas is that of the quanta of the electromagnetic field. We show that Bose–Einstein statistics delivers a very good model for these pieces of texts telling stories, both for short stories and for long stories of the size of novels. We analyze an unexpected connection with Zipf’s law in human language, the Zipf ranking relating to the energy levels of the words, and the Bose–Einstein graph coinciding with the Zipf graph. We investigate the issue of ‘identity and indistinguishability’ from this new perspective and conjecture that the way one can easily understand how two of ‘the same concepts’ are ‘absolutely identical and indistinguishable’ in human language is also the way in which quantum particles are absolutely identical and indistinguishable in physical reality, providing in this way new evidence for our conceptuality interpretation of quantum theory. (shrink)

We model a piece of text of human language telling a story by means of the quantum structure describing a Bose gas in a state close to a Bose–Einstein condensate near absolute zero temperature. For this we introduce energy levels for the words used in the story and we also introduce the new notion of ‘cogniton’ as the quantum of human thought. Words are then cognitons in different energy states as it is the case for photons in different energy states, (...) or states of different radiative frequency, when the considered boson gas is that of the quanta of the electromagnetic field. We show that Bose–Einstein statistics delivers a very good model for these pieces of texts telling stories, both for short stories and for long stories of the size of novels. We analyze an unexpected connection with Zipf’s law in human language, the Zipf ranking relating to the energy levels of the words, and the Bose–Einstein graph coinciding with the Zipf graph. We investigate the issue of ‘identity and indistinguishability’ from this new perspective and conjecture that the way one can easily understand how two of ‘the same concepts’ are ‘absolutely identical and indistinguishable’ in human language is also the way in which quantum particles are absolutely identical and indistinguishable in physical reality, providing in this way new evidence for our conceptuality interpretation of quantum theory. (shrink)

This book is devoted to the study of human thought, its systemic structure, and the historical development of mathematics both as a product of thought and as a fascinating case analysis. After demonstrating that systems research constitutes the second dimension of modern science, the monograph discusses the yoyo model, a recent ground-breaking development of systems research, which has brought forward revolutionary applications of systems research in various areas of the traditional disciplines, the first dimension of science. After the systemic structure (...) of thought is factually revealed, mathematics, as a product of thought, is analyzed by using the age-old concepts of actual and potential infinities. In an attempt to rebuild the system of mathematics, this volume first provides a new look at some of the most important paradoxes, which have played a crucial role in the development of mathematics, in proving what these paradoxes really entail. Attention is then turned to constructing the logical foundation of two different systems of mathematics, one assuming that actual infinity is different than potential infinity, and the other that these infinities are the same. This volume will be of interest to academic researchers, students and professionals in the areas of systems science, mathematics, philosophy of mathematics, and philosophy of science. (shrink)

The article examines the current state of the mathematical education of the students-philosophers that depends on language of the humanitarian mathematics, evidence of its statements and methodological problem of the cognition of the mathematical facts. One of important tasks of philosophy of mathematical education consists in motivation of the need for training mathematics of students-philosophers. The main criterion of the usefulness of mathematics for philosophers is revealed in the ways of justification of its truth and completeness of reasoning of mathematical (...) statements. This reflects the attractiveness of mathematical knowledge for philosophers that is characterized by the fact that the truth is revealed along with a proof in mathematics. One of the main so-called ‘acupuncture points‘ of mathematical education of philosophers, as the author believes, is the language of modern mathematics. In General, the linguistic aspect is very important in modern science. The second ‘acupuncture point‘ of mathematical education of philosophers is the level of evidence claims. Third, according to the author, is responsible for knowledge of mathematical truths, is involved in not only our existence, but also transcendental knowledge. In the context of philosophy of the future of modern civilization, mathematics is indispensable as a necessary element of scientific and philosophical picture of the world. To support the conclusions the author draws a broad philosophical and scientific context. (shrink)

0. 0 Psychology versus Complex Systems Science Over the last century, psychology has become much less of an art and much more of a science. Philosophical speculation is out; data collection is in. In many ways this has been a very positive trend. Cognitive science (Mandler, 1985) has given us scientific analyses of a variety of intelligent behaviors: short-term memory, language processing, vision processing, etc. And thanks to molecular psychology (Franklin, 1985), we now have a rudimentary understanding of the chemical (...) processes underlying personality and mental illness. However, there is a growing feeling-particularly among non-psychologists (see e. g. Sommerhoff, 1990) - that, with the new emphasis on data collection, something important has been lost. Very little attention is paid to the question of how it all fits together. The early psychologists, and the classical philosophers of mind, were concerned with the general nature of mentality as much as with the mechanisms underlying specific phenomena. But the new, scientific psychology has made disappointingly little progress toward the resolution of these more general questions. One way to deal with this complaint is to dismiss the questions themselves. After all, one might argue, a scientific psychology cannot be expected to deal with fuzzy philosophical questions that probably have little empirical signifi cance. It is interesting that behaviorists and cognitive scientists tend to be in agreement regarding the question of the overall structure of the mind. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is (...) a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

What is a language? What do scientific grammars tell us about the structure of individual languages and human language in general? What kind of science is linguistics? These and other questions are the subject of Ryan M. Nefdt's Language, Science, and Structure. -/- Linguistics presents a unique and challenging subject matter for the philosophy of science. As a special science, its formalisation and naturalisation inspired what many consider to be a scientific revolution in the study of mind and language. Yet (...) radical internal theory change, multiple competing frameworks, and issues of modelling and realism have largely gone unaddressed in the field. Nefdt develops a structural realist perspective on the philosophy of linguistics which aims to confront the aforementioned topics in new ways while expanding the outlook toward new scientific connections and novel philosophical insights. On this view, languages are real patterns which emerge from complex biological systems. Nefdt's exploration of this novel view will be especially valuable to those working in formal and computational linguistics, cognitive science, and the philosophies of science, mathematics, and language. (shrink)

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...) mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from (...) each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system. (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)

Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

The article is devoted to aesthetic nature of the philosophy of dance as a rapidly developing area of studying. The aesthetic issues of choreographies in the cognitive context have not been properly studied. The mathematical component of the classical ballet, which is shown through the internal patterns of the expressiveness of the different types of dance movements in the system of artistic thinking, is analyzed in a wide range of the philosophical problems of art of dancing. The substantial triad of (...) structure of dance is considered at the level of philosophical research. In it, along with static and dynamic forms of dance, the third component, which reflects physicality, related to the ‘sense of movement‘ or the perception of information through body movements is added to the binary structure of dance. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...) mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess anexactquantity of the elements in a (...) collection without counting them), or on the ability to approximate quantities (i.e., to assess anapproximatequantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics. (shrink)

Quantum Structures and the Nature of Reality is a collection of papers written for an interdisciplinary audience about the quantum structure research within the International Quantum Structures Association. The advent of quantum mechanics has changed our scientific worldview in a fundamental way. Many popular and semi-popular books have been published about the paradoxical aspects of quantum mechanics. Usually, however, these reflections find their origin in the standard views on quantum mechanics, most of all the wave-particle duality picture. Contrary to (...) relativity theory, where the meaning of its revolutionary ideas was linked from the start with deep structural changes in the geometrical nature of our world, the deep structural changes about the nature of our reality that are indicated by quantum mechanics cannot be traced within the standard formulation. The study of the structure of quantum theory, its logical content, its axiomatic foundation, has been motivated primarily by the search for their structural changes. Due to the high mathematical sophistication of this quantum structure research, no books have been published which try to explain the recent results for an interdisciplinary audience. This book tries to fill this gap by collecting contributions from some of the main researchers in the field. They reveal the steps that have been taken towards a deeper structural understanding of quantum theory. (shrink)

Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of mathematics in science (...) is an important trigger for the objectivity of mathematical knowledge. (shrink)

Investigation into the sequence structure of the genetic code by means of an informatic approach is a real success story. The features of human language are also the object of investigation within the realm of formal language theories. They focus on the common rules of a universal grammar that lies behind all languages and determine generation of syntactic structures. This universal grammar is a depiction of material reality, i.e., the hidden logical order of things and its relations determined by natural (...) laws. Therefore mathematics is viewed not only as an appropriate tool to investigate human language and genetic code structures through computer sciencebased formal language theory but is itself a depiction of material reality. This confusion between language as a scientific tool to describe observations/experiences within cognitive constructed models and formal language as a direct depiction of material reality occurs not only in current approaches but was the central focus of the philosophy of science debate in the twentieth century, with rather unexpected results. This article recalls these results and their implications for more recent mathematical approaches that also attempt to explain the evolution of human language. (shrink)

I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...) of mathematics. Further, it makes explicit how the brand of non-representationalism I develop is compatible with there being substantive rationality constraints on our mathematical theorizing. (shrink)

Many mathematicians are platonists: they believe that the axioms of mathematics are true because they express the structure of a nonspatiotemporal, mind independent, realm. But platonism is plagued by a philosophical worry: it is unclear how we could have knowledge of an abstract, realm, unclear how nonspatiotemporal objects could causally affect our spatiotemporal cognitive faculties. Here I aim to make room in our metaphysical picture of the world for the causal relevance of abstracta.

Guided by key insights of the four great philosophers mentioned in the title, here, in review of and expanding on our earlier work (Burchard, 2005, 2011), we present an exposition of the role played by language, & in the broader sense, λογοζ, the Logos, in how the CNS, the brain, is running the human being. Evolution by neural Darwinism has been forcing the linguistic nature of mind, enabling it to overcome & exploit the cognitive gap between an animal and its (...) world by recognizing environmental structures. Our work was greatly influenced by Heidegger’s lecture notes on metaphysics (Heidegger, 1935). We found agreement with recent progress in neuroscience, but also mathematical foundations of language theory, equating Logos with the mathematical concept of structure. The mystery of perception across the gap is analyzed as radiation and molecules impinging on sensory neurons that carry linguistic information about gross environmental structures, and only remotely about the physical reality of elementary particles. The most important logical brain function is Ego or Self, guiding the workings of the brain as a logos machine. Ego or Self operates from neurons in frontopolar cortex with global receptive fields. The logos machine can function only by availing itself of global context, its internally stored noumenal cosmos NK, and the categorical-conceptual apparatus CCA, updated continually through the neural default mode network (Raichle, 2005). In the Transcendental Deduction, Immanuel Kant discovered that Ego or Self is responsible for conscious control in perception relying on concepts & categories for a fitting percept to be incorporated intoNK. The entire CNS runs as a “movie-in-the-brain” (Parvizi & Damasio, 2001), at peak speed processing simultaneously in a series of cortical centers a stack of up to twelve frames in gamma rhythm of 25 ms intervals. We equate global context, or NK, with our human world, Heidegger’s Dasein being-in-the-world, and are able to demonstrate that the great philosopher in EM parallels neuro-science concerning the human mind. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- Blending Knowledge Structures in the (...) Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents (...) and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself. (shrink)

The ancient problem of the relationship of the continuous to the discrete, since its discovery by the Greeks, has posed a range of immensely fruitful challenges to both philosophical and mathematical thought, leading to a variety of mathematical and conceptual innovations whose positive development actively continues today. In this brief section introduction, I selectively outline some significant moments at which this problem has provided important historical occasions for concrete mathematical innovation as well as closely linked philosophical insights, before introducing the (...) particular contributions of the section authors to historical and contemporary research and practice into the nature and structure of the continuum, the construction of real numbers, the foundations of analysis, and the logical, set-theoretical, and computational underpinnings of the structure of continuously valued numbers in the most general sense, in both constructive and nonconstructivist contexts. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...) is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

Within the philosophy of science, the realism debate has been revitalised by the development of forms of structural realism. These urge a shift in focus from the object oriented ontologies that come and go through the history of science to the structures that remain through theory change. Such views have typically been elaborated in the context of theories of physics and are motivated by, first of all, the presence within such theories of mathematical equations that allow straightforward representation of the (...) relevant structures; and secondly, the implications of such theories for the individuality and identity of putative objects. My aim in this talk is to explore the possibility of extending such views to biological theories. An obvious concern is that within the context of the latter it is typically insisted that we cannot find the kinds of highly mathematised structures that structural realism can point to in physics. I shall indicate how the model-theoretic approach to theories might help allay such concerns. Furthermore, issues of identity and individuality also arise within biology. Thus Dupre has recently noted that there exists a ‘General Problem of Biological Individuality’ which relates to the issue of how one divides ‘massively integrated and interconnected’ systems into discrete components. In response Dupre advocates a form of ‘Promiscuous Realism’ that holds, for example, that there is no unique way of dividing the phylogenetic tree into kinds. Instead I shall urge serious consideration of those aspects of the work of Dupre and others that lean towards a structuralist interpretation. By doing so I hope to suggest possible ways in which a structuralist stance might be extended to biology. (shrink)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and intuition of (...) geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and intuition of (...) geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for a measure of (...) collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate. (shrink)

The structure-mapping theory has become the de-facto standard account of analogies in cognitive science and philosophy of science. In this paper I propose a distinction between two kinds of domains and I show how the account of analogies based on structure-preserving mappings fails in certain (object-rich) domains, which are very common in mathematics, and how the axiomatic approach to analogies, which is based on a common linguistic description of the analogs in terms of laws or axioms, can be used successfully (...) to explicate analogies of this kind. Thus, the two accounts of analogies should be regarded as complementary, since each of them is adequate for explicating analogies that are drawn between different kinds of domains. In addition, I illustrate how the account of analogies based on axioms has also considerable practical advantages, e. g., for the discovery of new analogies. (shrink)

Mental representations remain the central posits of psychology after many decades of scrutiny. However, there is no consensus about the representational format(s) of biological cognition. This paper provides a survey of evidence from computational cognitive psychology, perceptual psychology, developmental psychology, comparative psychology, and social psychology, and concludes that one type of format that routinely crops up is the language-of-thought (LoT). We outline six core properties of LoTs: (i) discrete constituents; (ii) role-filler independence; (iii) predicate–argument structure; (iv) logical operators; (v) inferential (...) promiscuity; and (vi) abstract content. These properties cluster together throughout cognitive science. Bayesian computational modeling, compositional features of object perception, complex infant and animal reasoning, and automatic, intuitive cognition in adults all implicate LoT-like structures. Instead of regarding LoT as a relic of the previous century, researchers in cognitive science and philosophy-of-mind must take seriously the explanatory breadth of LoT-based architectures. We grant that the mind may harbor many formats and architectures, including iconic and associative structures as well as deep-neural-network-like architectures. However, as computational/representational approaches to the mind continue to advance, classical compositional symbolic structures – that is, LoTs – only prove more flexible and well-supported over time. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

It is shown how the historiographic purport of Lakatosian methodology of mathematics is structured on the theme of analysis and synthesis. This theme is explored and extended to the revolutionary phase around 1800. On the basis of this historical investigation it is argued that major innovations, crucial to the appraisal of mathematical progress, defy reconstruction as irreducibly rational processes and should instead essentially be understood as processes of social-cognitive interaction. A model of conceptual change is developed whose essential ingredients are (...) the variability of rational responses to new intellectual and practical challenges arising in the cultural environment of mathematics, and the shifting selective pressure of society. The resulting view of mathematical development is compared with Kuhn's theory of scientific paradigms in the light of some personal communications. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

This paper presents a speculative model of the cognitive mechanisms underlying the transition from episodic to mimetic (or memetic) culture with the arrival of Homo erectus, which Donald [1991] claims paved the way for the unique features of human culture. The model draws on Kauffman's [1993] theory of how an information-evolving system emerges through the formation of an autocatalytic network. Though originally formulated to explain the origin of life, this theory also provides a plausible account of how discrete episodic memories (...) become woven into an internal model of the world, or worldview, that both structures, and is structured by, self-triggered streams of thought. Social interaction plays a role in (and may be critical to) this process. Implications for cognitive development are explored. (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)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Handbook of Cognitive Mathematics ed. by Marcel DanesiNathan HaydonMarcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including indexFor one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit into the nature and study (...) of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to.The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it.This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice.The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry.A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical... (shrink)

This paper is a nontechnical exposition of the author's view that mathematics is a science of patterns and that mathematical objects are positions in patterns. the new elements in this paper are epistemological, i.e., first steps towards a postulational theory of the genesis of our knowledge of patterns.

The development of ancient civilizations and their achievements in sciences such as mathematics and astronomy are well researched for script-using civilizations. On the basis of oral tradition and mnemonic artifacts illiterate ancient civilizations were able to attain an adequate level of knowledge. The Neolithic and Bronze Age earthworks and circles are such mnemonic artifacts. Explanatory models are given for the shape of the stone formations and the ditch of Stonehenge reflecting the circular and specific non-circular shapes of these structures. The (...) basic mathematical concepts are Pythagorean triangles, thus adopting the construction procedures of the Neolithic circular ditches of Central Europe in the fifth Millennium bc and later earthworks and stone circles in Britain and Brittany. This knowledge was extended with new elliptical concepts. Approximations for the values of π and the square root of 2 are encoded in the henge. All constructions were performed using a standardized “Babylonian” metrology that shows a remarkable consistency and comprehensible development over some 14 centuries. (shrink)

The philosophy of science has lost its self-confidence. Structures in Science (2001) is an advanced textbook that explicates, updates and integrates the best insights of logical empiricism and its main critics. This "neo-classical approach" aims at providing heuristic patterns for research.The book introduces four ideal types of research programs (descriptive, explanatory, design and explicative) and reanimates the distinction between observational laws and proper theories without assuming a theory-free language. It explicates various patterns of explanation by subsumption and specification as well (...) as structures in reductive and other types of interlevel research. Its threefold analysis of theory evaluation leads to new characterizations of confirmation, empirical progress, and truth approximation. What emerges are partial analogies between progress in nomological research, presented in detail in From Instrumentalism to Constructive Realism (2000) and progress in explicative and design research. Finally, special chapters are devoted to design research programs, computational philosophy of science, the structuralist approach to theories, and research ethics.The present synopsis of Structures in Science highlights the main topics, the final emphasis being on design research and research ethics. (shrink)