We focus on the thesis that action understanding is a function of the mirror neuron system. According to our opinion, understanding is a process that runs through hermeneutic circles from the “Vorverständnis” (“previous understanding”) to steps of deeper understanding. Our critique relates to the narrow neuroscientific definition of action understanding as the capacity to recognize several movements as belonging to one action. After a reconstruction of the model's developments, we will challenge the claims of the model by Rizzolatti and Sinigaglia (...) (2010). By analyzing the relation between the experimental results and its interpretation, we will conclude that there is no proof that mirror neuron activity leads to action understanding. (shrink)

__Language Proof and Logic_ is available as a physical book with the software included on CD and as a downloadable package of software plus the book in PDF format. The all-electronic version is available from Openproof at ggweb.stanford.edu._ The textbook/software package covers first-order language in a method appropriate for first and second courses in logic. An on-line grading services instantly grades solutions to hundred of computer exercises. It is designed to be used by philosophy instructors teaching a logic course (...) to undergraduates in philosophy, computer science, mathematics, and linguistics. Introductory material is presented in a systematic and accessible fashion. Advanced chapters include proofs of soundness and completeness for propositional and predicate logic, as well as an accessible sketch of Godel's first incompleteness theorem. The book is appropriate for a wide range of courses, from first logic courses for undergraduates to a first graduate logic course. The software package includes four programs: Tarski's World 5.0, a new version of the popular program that teaches the basic first-order language and its semantics; Fitch, a natural deduction proof environment for giving and checking first-order proofs; Boole, a program that facilitates the construction and checking of truth tables and related notions ; Submit, a program that allows students to submit exercises done with the above programs to the Grade Grinder, the automatic grading service. Grade reports are returned to the student and, if requested, to the student's instructor, eliminating the need for tedious checking of homework. All programs are available for Windows, Macintosh and Linux systems. Instructors do not need to use the programs themselves in order to be able to take advantage of their pedagogical value. More about the software can be found at lpl.stanford.edu. The price of a new text/software package includes one Registration ID, which must be used each time work is submitted to the grading service. Once activated, the Registration ID is not transferable. (shrink)

Mathematical proofs are not sequences of arbitrary deductive steps—each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities—that is, sequences of deductive inferences—and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms of (...) rational planning agency, and that a deductive step in a mathematical proof qualifies as rational whenever the corresponding deductive inference in the associated proof activity figures in a plan that has been constructed rationally. It is then shown that mathematical proofs whose associated proof activities violate these norms are likely to be judged as defective by mathematical agents, thereby providing evidence that these norms are indeed present in mathematical practice. We conclude that, if mathematical proofs are not mere sequences of deductive steps, if they possess a rational structure, it is because they are the product of rational planning agents. (shrink)

Prawitz has recently developed a theory of epistemic grounding that differs in many respects from his earlier semantics of arguments and proofs. An innovative approach to inferences yields a new conception of the intertwinement of the notions of valid inference and proof. We aim at singling out three reasons that may have led Prawitz to the ground-theoretic turn, i.e.: a better order in the explanation of the relation between valid inferences and proofs; a notion of valid inference based on (...) which valid inferences and proofs are recognisable as such; a reconstruction of the deductive activity that makes inferences capable of yielding justification per se. These topics are discussed by Prawitz with reference to a very general and ancient question: why and how correct deduction has the epistemic power to compel us to accept its conclusions, provided its premises are justified? We conclude by remarking that, in spite of some improvements, the ground-theoretic approach shares with the previous one a problem of vacuous validity which, as Prawitz himself points out, blocks in both cases a satisfactory explanation of epistemic compulsion. (shrink)

We present a setting in which the search for a proof of B or a refutation of B can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates (...) to a refutation of the formula. The game is described for multiplicative and additive linear logic . A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win. (shrink)

Responsible Research and Innovation (RRI) is described as a new way of doing science that brings science closer to society. Based on a qualitatively oriented case study, this article supports previous research indicating that researchers face a variety of ethical problems and dilemmas when implementing RRI for the first time. These include difficulties with anticipating and controlling future impacts, an asymmetry of power between project partners and an elusive understanding of the RRI concept. The researchers’ challenges were rooted in conventional (...) research ethics and could be boiled down to one core dilemma: If RRI had been applied from the very beginning of the project period, the chance of realising proof of concept within the scheduled time may decrease. The researchers’ solution to this dilemma was to prioritize proof of concept and postpone RRI activities to later stages of the project. If RRI is expected to live up to its ambition of representing a new way of doing science, more effort is needed at the political level to facilitate change. (shrink)

This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will (...) be essential reading for all those working in logic & computer science. (shrink)

This textbook/software package covers first-order language in a method appropriate for first and second courses in logic. The unique on-line grading services instantly grades solutions to hundred of computer exercises. It is specially devised to be used by philosophy instructors in a way that is useful to undergraduates of philosophy, computer science, mathematics, and linguistics. The book is a completely rewritten and much improved version of The Language of First-order Logic. Introductory material is presented in a more systematic and accessible (...) fashion. Advanced chapters include proofs of soundness and completeness for propositional and predicate logic, as well as an accessible sketch of Godel's first incompleteness theorem. The book is appropriate for a wide range of courses, from first logic courses for undergraduates to a first graduate logic course. The package includes four pieces of software: Tarski's World 5.0, a new version of the popular program that teaches the basic first-order language and its semantics; Fitch, a natural deduction proof environment for giving and checking first-order proofs; Boole, a program that facilitates the construction and checking of truth tables and related notions ; Submit, a program that allows students to submit exercises done with the above programs to the Grade Grinder, the automatic grading service. Grade reports are returned to the student and, if requested, to the student's instructor, eliminating the need for tedious checking of homework. All programs are available for Windows, Macintosh and Linux systems. Instructors do not need to use the programs themselves in order to be able to take advantage of their pedagogical value. More about the software can be found at lpl.stanford.edu. The price of a new text/software package includes one Registration ID, which must be used each time work is submitted to the grading service. Once activated, the Registration ID is not transferable. (shrink)

The past twenty years have witnessed extensive collaborative research between computer scientists, logicians, linguists, philosophers, and psychologists. These interdisciplinary studies stem from the realization that researchers drawn from all fields are studying the same problem. Specifically, a common concern amongst researchers today is how logic sheds light on the nature of information. Ancient questions concerning how humans communicate, reason and decide, and modern questions about how computers should communicate, reason and decide are of prime interest to researchers in various disciplines. (...) _Words, Proofs and Diagrams_ is a collection of papers covering active research areas at the interface of logic, computer science, and linguistics. Readers of the volume will find traditional research on process logics, issues in formal semantics, and language processing. In addition, the volume also highlights a particularly new area where all three disciplines meet—the study of images and graphics as information carriers and the diagrammatic reasoning supported by them. (shrink)

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on (...) the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour. (shrink)

“Proof of concept” is a phrase frequently used in descriptions of research sought in program announcements, in experimental studies, and in the marketing of new technologies. It is often coupled with either a short definition or none at all, its meaning assumed to be fully understood. This is problematic. As a phrase with potential implications for research and technology, its assumed meaning requires some analysis to avoid it becoming a descriptive category that refers to all things scientifically exciting. I (...) provide a short analysis of proof of concept research and offer an example of it within synthetic biology. I suggest that not only are there activities that circumscribe new epistemological categories but there are also associated normative ethical categories or principles linked to the research. I examine these and provide an outline for an alternative ethical account to describe these activities that I refer to as “extended agency ethics”. This view is used to explain how the type of research described as proof of concept also provides an attendant proof of principle that is the result of decision-making that extends across practitioners, their tools, techniques, and the problem solving activities of other research groups. (shrink)

This book's activities highlight the important cycle of exploration, conjecture, and justification in all five mathematical strands. Students recognize patterns and make conjectures, learn the value of a counterexample, explore the strengths and weaknesses of visual proofs, discover the power of algebraic representations, and learn that theoretical approaches can substantiate empirical results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. --publisher description.

This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments (...) and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion. (shrink)

Planning with incomplete knowledge becomes a very active research area since late 1990s. Many logical formalisms introduce sensing actions and conditional plans to address the problem. The action language $\mathcal{A}_{K}$ invented by Son and Baral is a well-known framework for this purpose. In this paper, we propose so-called cautious and weakly cautious semantics for $\mathcal{A}_{K}$ , in order to allow an agent to generate and execute reliable plans in safety-critical environments. Intuitively speaking, cautious and weakly cautious semantics enable the agent (...) to know exactly what happens after the execution of an action. Computational complexity analysis shows that cautious semantics reduces the reasoning complexity of $\mathcal{A}_{K}$ , it is also worth to point out that many useful domains could still be expressed with this setting. Another important contribution of our work is the development of Hoare style proof systems. These proof systems are served as inference mechanisms for the verification of conditional plans, and proved to be sound and complete. In addition, they could also be used for plan generation, in the sense that constructing a derivation is indeed a procedure to finding a plan. We point out that the proof systems posses a nice property for off-line planning, that is, the agent could generate and store short proofs in her spare time, and perform quick plan query by easily constructing a long proof from the stored shorter ones (under the assumption that sufficient proofs are stored). (shrink)

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he (...) used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...) that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)

In his Grundgesetze, §32, Frege launched the idea that the meaning of a sentence is given by its truth condition, or, in his particular version, the condition under which it will be a name of the True. This, indeed, was only one of the many roles in which truth has to serve within the Fregean system. In particular, truth is an absolute notion in the sense that bivalence holds: every Gedanke is either true or false, in complete independence of any (...) conative activity, whether by God or man. Thus various epistemological notions, such as the correctness of an assertion made, or judgement passed, are reducible to this absolute notion of truth: an assertion made through the utterance of a declarative sentence is correct when the proposition expressed by the sentence in question is true. Given this absolute status of truth it is not surprising that Frege is of the opinion that truth is a sui generis notion which has to be left unanalyzed and, indeed, which is indefinable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Feminist Studies 41, no. 3. © 2015 by Feminist Studies, Inc. 623 Jaye Cee Whitehead, Kath Bassett, Leia Franchini, and Michael Iacolucci “The Proof Is in the Pudding”: How Mental Health Practitioners View the Power of “Sex Hormones” in the Process of Transition In the United States today, popular discourse touts the power of “sex hormones” and hormone receptors in the brain to chemically produce gender expressions (manifested (...) in physical sex traits, behaviors, and attitudes) and identities (a sense of self as feminine or masculine). These accounts range from common assumptions that Western ideals of femininity (such as empathy understanding, cooperative orientation, nurturing behavior) and masculinity (rational/spatial understanding, competitive orientation, violent behavior) are produced by corresponding sex hormones, to studies that reduce complex gender identities and macrosocial economic phenomena to presocial, binary, hormonal processes.1 For sociologists, accounts that reduce complex identities to internal hormonal reactions are problematic for two primary reasons: first, they are inaccurate because they fail to examine how hormonal reactions are embedded in a complex cultural process of meaning and interpretation; second, they operate as bioreductive ideologies that obscure the social power inherent to the process of gender identity construction. In what follows, we extend both of these insights by examining mental health 1. For an example of a hybrid popular-scientific account of the power of sex hormones to destabilize economic structures, see Jon Coates and Joe Herbert, “Endogenous Steroids and Financial Risk Taking on a London Trading Floor,” Proceedings of the National Academy of Sciences 105, no. 16 (April 2008): 6167–72. 624 Whitehead, Bassett, Franchini, and Iacolucci practitioners’ divergent interpretations of the power of sex hormones in the process of gender transition.2 In doing so, we examine the cultural power of sex hormones in crafting gender identities without reproducing reductionist interpretations of trans embodiment. Perhaps what is most perplexing about the way we speak of sex hormones is that we know it is inaccurate to describe these hormones as sexed; scientists have acknowledged since the 1930s that they are neither sex specific in their function, nor in terms of their location in male or female bodies.3 In her extensive historicization of sex hormones, Anne Fausto-Sterling concludes that it is more accurate to call androgen- and estrogen-based hormones, “steroid hormones” as they have functions that are not confined to corresponding sexed bodies. In fact she urges scientists to “break out of the sex hormone straightjacket” and to look at steroids as just one of a number of components that are important to the creation of sex and gender, including environment and experience.4 Scholars also point to an inextricable link between the chemical operation of hormones and the social process of constructing meaning, both at the level of social interaction and macrocultural constructions of sex categories and gender ideologies.5 While brain organization/activation theory attributes sex and gender differences to hormonal interactions within the developing brain, Fausto-Sterling questions the distinction between activation and organization by pointing out how “the brain can respond to hormonal stimuli with anatomical changes…hormonal systems, after all, respond exquisitely to experience, be it in the form of nutrition, stress, or sexual activity (to name but a few possibilities ).”6 Drawing from Elizabeth Grosz, Fausto-Sterling argues that the power of hormones is best understood according to the model of the 2. “Gender transition” is a controversial phrase, as it implies that transgendered individuals who receive hormone therapy are changing identities. We are only using it for lack of clear alternative phrasing, not because we believe body modifications change gender identities. 3. See Anne Fausto-Sterling, Sexing the Body: Gender Politics and the Construction of Sexuality (New York: Basic Books, 2000); Rebecca Jordan-Young, Brainstorm: The Flaws in the Science of Sex Differences (Cambridge, MA: Harvard University Press, 2010); and Nelly Oudshoorn, Beyond the Natural Body: An Archeology of Sex Hormones (New York: Routledge, 1994). 4. Fausto-Sterling, Sexing the Body, 193–94. 5. Fausto-Sterling, Sexing the Body. 6. Ibid., 232. Whitehead, Bassett, Franchini, and Iacolucci 625 Möbius strip, wherein internal components of the self are always connected to and continuous with outer components such as culture and environment.7 In other... (shrink)

More or less explicitly inspired by the Aristotelian classification of arguments, a wide tradition makes a sharp distinction between argument and proof. Ch. Perelman and R. Johnson, among others, share this view based on the principle that the conclusion of an argument is uncertain while the conclusion of a proof is certain. Producing proof is certainly a major part of mathematical activity. Yet, in practice, mathematicians, expert or beginner, argue about mathematical proofs. This happens during the search (...) for a proof, then when the proof is presented and discussed by experts, and finally when it is taught or used in didactical contexts. (shrink)

An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number theory, and (...) more. The exposition guides and mentors the reader through an adventure in mathematical discovery, requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Ultimately, this is really a book about productive struggle and learning how to learn. This is a print version of the popular open-access online text by Dana C. Ernst. (shrink)

According to Iris Murdoch, the chief experience in morality is the recognition of others, and this is the experience of loving attention. Love is an independent source of moral authority, distinct from the authority of reason. It is independent because it can be attained through moral experiences that are not certified by reason and cannot be achieved by rational deliberation. This view of love calls into question a cluster of concepts, such as rational agency and principled action, which figure prominently (...) in the Kantian characterization of the moral experience and resound through Kant’s account of virtue as an endured but never-ending struggle. But it would be a mistake to conclude that Murdoch’s philosophical project opposes Kant’s without further qualifications. In this chapter, I illustrate some convergence between Kant and Murdoch at the methodological level, in their polemic against reductivism. They both oppose reductivist empiricism on epistemological and moral grounds, because they define the standard of proof in relation to the experience of morality. In Murdoch’s words, the philosophical proof, if there is one, is the same as the moral proof (Murdoch 1997: 361). This is where Kant and Murdoch part ways. While Kant argues that the moral experience is the experience of the impact of reason, which provides self-discipline by constraint, Murdoch holds that the moral standard of proof is love. The transformative power of love allows us to engage with reality and recognize others for what they are, that in which consists the primal moral experience. Murdoch’s argument against Kant’s appeal to principles brings to the fore a crucial issue, which concerns the source of moral authority and the possibility of practical reason. Some of her most popular arguments against Kant are based on an oversimplified view of the impact of reason in moral life, and perhaps they owe part of their fortune to this oversimplification. I shall argue that such arguments miss their intended target, but I also hope to show that Murdoch sketches a distinctive model of the moral proof, which establishes a genuine alternative both to theories of practical reason and to reductivist accounts of the mind and its activities. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. (...) The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

In this paper the class of minimal models C ZI for Kiczuk's system of physical change ZI is provided and soundness and completeness proofs of ZI with respect to these models are given. ZI logic consists of propositional logic von Wright's And Then and six specific axioms characterizing the meaning of unary propositional operator "Zm", read "there is a change in the fact that". ZI is intended to be a logic which provides a formal account for describing two kinds of (...) process change: the change from one state of the process to its other state (e.g., transmitting or absorbing energy with greater or less than the usual intensity) and the perishing of the process (e.g., cessation of the energetic activity of the sun). (shrink)

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

An exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice (...) of mathematics. Inspired by the experiments of the Paris-based writing group known as the Oulipo—whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp—Ording explores new ways to examine the aesthetic possibilities of mathematical activity. 99 Variations on a Proof is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor. Readers will gain not only a bird’s-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Nature of Proof in PsychiatryPaul Lieberman (bio)Keywordspsychotherapy process, knowledge and psychiatry, externalism, WittgensteinThis vivid clinical report illustrates recognizably, and provocatively, a number of routine, but often unexamined, clinical questions. In its few paragraphs, it depicts challenges that each practitioner confronts, and, in the flux of clinical work, addresses, however implicitly and imperfectly, every day: From what data, and by what processes, does a clinical formulation, or way (...) of understanding, come into being? How do we appraise its accuracy and adequacy? Of what value is even a well-substantiated understanding, in the mind of the therapist, to the patient? What is the power of empathy as a treatment tool? How do we assess someone's status as a “good candidate for therapy?” When, typically, only the outlines of a problem are visible, and its origins and implications are largely hidden, by what right do we recommend psychotherapy, and with what conception, at what level of detail, of what its outcome will be? These are among the questions that Dr. Bailey's story challenges us to address.They are clinical questions, of course, because they deal with empirical matters of fact—psychological processes, outcomes studies, and evidence. But they are also philosophical because, as Freud insisted and Dr. Bailey shows, information obtained in the clinical situation often has power and authority that lead us to believe we understand what is going on, understand the other person, even in the absence of scientifically tested hypotheses, or the clear application of research-based evidence. Interactions between people—including clinical interactions—have a life and ability to engage us. Their meanings can seem so transparent and convincing that no doubt or questioning may seem cogent. At their most powerful, such experiences can reawaken us to important realms of experience and feeling which we might, otherwise, forget, ignore, suppress, deny, or in other ways keep out of awareness; they have the potential to arouse us and reconnect us with truths that, once revived, can seem both valid and important—perhaps the kind of experience that we may derive from works of art or literature (in fact, Dr. Bailey's account could be seen as a clinical parable—meant to be neither true nor false, but instructive). But even less dramatic examples are commonplace: we do routinely claim to know and be able to interpret and understand the actions of other people from our everyday interactions with and observations of them.What is less clear is how to account for these convictions and how to appraise their value, because they do not derive from established scientific procedures. Clinical intuition has a role throughout medicine, but we usually expect it to be subjected later to more rigorous tests to establish the correct diagnosis. Yet, as Dr. Bailey's case illustrates, clinical intuition in the psychiatric encounter seems somehow different. When Colin [End Page 225] leads Dr. Bailey to the conclusion that a “painful lack of wounds” has made him depressed, we find this credible. It seems similarly credible that Dr. Bailey might be anxious about his own capacity to defend his young family, because he says he is, that he perceives Colin as testing his manhood or sincerity, because that seems to fit Colin's temperament, that being accurately understood is helpful, because most of us have had that experience and valued it, and that, most generally, an absence of challenge or of activity could be depressing. But suppose a patient believed that a painful lack of wounds produced diabetes instead of depression, or suppose he had developed epilepsy and attributed that to an absence of combat experience. We might see these as ‘pre-scientific’ or ‘hypothesis-generating’ (or worse). By contrast, it seems timid to consider psychiatric formulations to be always and only provisional or hypothetical.In what ways and to what extent should we expect psychiatry to be different, and why? Several considerations suggest that psychiatry's reasons are not, always and everywhere, the same as those of general medicine, nor are its objectives.In general medicine, we distinguish between signs and symptoms, on the one hand, and “underlying” disease processes, on the other. But in psychiatry, this distinction can be... (shrink)

This paper explains a way of understanding Kant's proof of God's existence in the Critique of Practical Reason that has hitherto gone unnoticed and argues that this interpretation possesses several advantages over its rivals. By first looking at examples where Kant indicates the role that faith plays in moral life and then reconstructing the proof of the second Critique with this in view, I argue that, for Kant, we must adopt a certain conception of the highest good, and (...) so also must choose to believe in the kind of God that can make it possible, because this is essentially a way of actively striving for virtue. One advantage of this interpretation, I argue, is that it is able to make sense of the strong link Kant draws between morality and religion. (shrink)

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged (...) it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians. (shrink)

After presenting the “first and more manifest way” of proving the existence of God by reason alone in Summa Theologiae Ia, 2, 3, Saint Thomas Aquinas continues this project by turning in the “Second Way” to what he somewhat enigmatically calls “the nature of the efficient cause.” The greatest obstacle to understanding his Second Way, though, is determining precisely what Aquinas means by “the nature of the efficient cause” and “an order of efficient causes,” and how the Second Way is (...) distinct from the First and Third Ways. This paper shows that instead of purporting to demonstrate God's existence as the cause of metaphysical being, as is commonly supposed, Aquinas is actually searching for the ultimate cause of immanent activity in rational beings. It argues for this reading both from textual evidence internal to the Second Way and in the First Way. (shrink)

The anti-tuberculosis BCG vaccine was conceived and developed between 1905 and 1921 at Pasteur Institutes in France. Between 1921 and A. Calmette’s death in 1933, the vaccine went through a first period of national and international production and distribution for its use in humans. In France these activities were exclusively carried out by Calmette and his collaborators at the Pasteur Institute in Paris. Initially improvised production in a small room in the cellar gave way in 1931 to the construction (...) of the spacious and magnificent ‘New laboratories for research on tuberculosis and the preparation of the BCG’ within the premises of the Pasteur Institute. Presentation and image-building of the vaccine in France insisted on the fact that the BCG was not a commercial specialty but distributed free of charge. The technical monopoly of its production nevertheless lay with the Paris Pasteur Institute and standardization of scientific proof of safety, efficacy and stability was dominated by that Institute in France. In contrast, the international production and distribution of the vaccine was entrusted and transferred, free of charge, to trustworthy laboratories outside France. Multiplication of producers and users led to an increased need for standardization. For this process the analysis distinguishes between the standardization of scientific proof concerning safety, efficacy and stability of the vaccine and standardization of its medical uses. Whereas standardization was rather successful in the inter-war period in France, the international efforts remained rather unsuccessful. Only after world war II under Scandinavian leadership and in the context of mass vaccination programs supported by the WHO and UNICEF was the international standardization effectively implemented and succeeded at least to some extend. (shrink)

We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this (...) article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant. (shrink)

This book is a thoroughgoing analysis, interpretation, and defense of John Stuart Mill's proof of the principle of utility. It answers the traditional charges levelled against that proof, supports a comprehensive interpretation by painstaking study of Mill's text in Utilitarianism , and marshals arguments on behalf of utility as the first principle of morality. Universal Justice is dedicated to the advancement of justice conceived globally. It publishes interpretations of the history of thought as well as original monographs and (...) collective volumes, including work related to the activities of the International Society for Universalism. (shrink)

We examine the notion of inquiry and argue that philosophic inquiry is a transcendental activity. Activities, viewed as conforming to intelligible canons, applying to appropriate contexts, and directed to specifiable ends, are contrasted with their empirical descriptions. Inquiry, characterized as an internalized, continuous activity directed to an intrinsic end, and fundamentally presupposed by other activities, is considered at the levels of (1) science, (2) philosophy and (3) transcendental philosophy. We argue that (2) is a transcendental activity which determines (...) non-empirical concepts and is presupposed by (1). Alternative philosophic frameworks are grounded on hypothetical canons conceived by intelligence itself, which imply interpretations of objectivity and universality claiming validity for the community of inquirers, but they can always be rejected. We consider the possibility of categorical canons operating as second-order rules necessarily presupposed for the formulation of alternative philosophic frameworks, and (3) would be the activity of identifying, justifying and applying such transcendental canons of inquiry in general. Finally, we suggest that the only possible justification of such categorical canons would be a kind of ontological proof. Thus, a given philosophic approach reflects the transcendental activity of determining non-empirical concepts through the implementation of its fundamental regulative norms; but transcendental philosophy would determine the concept of philosophy itself through implementation of the categorical canons of inquiry in the construction of philosophic systems. (shrink)

THE CONCEPTS OF ASYMMETRY and optical activity, although introduced fairly late into inorganic chemistry, have played venerable and central roles in organic chemistry. Modern organic chemistry is usually considered to commence with Friedrich Wohler's synthesis of urea in 1828, and Jean Baptiste Biot's discovery of optical activity in 1812 antedates the very genesis of this field. Moreover, Joseph Achille Le Bel and Jacobus Henricusvan't Hoff's concept of the tetrahedral carbon atom, which constitutes the foundation of stereochemistry, was proposed a century (...) ago (1874), primarily in order to explain the optical isomerism investigated by Louis Pasteur and others. It is to Alfred Werner, the founder of coordination chemistry, however, that we owe the introduction of the concept of optical activity into coordination chemistry, together with its experimental proof. ' The purpose of this paper is to examine and speculate upon the circumstances of Werner's discovery of optically active coordination compounds, to trace its antecedents in earlier work, and to assess its impact upon the future course of inorganic stereochemistry. (shrink)

While recent studies have shed some light on the significance of the electrical activity of the nervous system, there has been no adequate explanation for the wave formation or synchronization of this electrical activity. Adrian sums up the problem. “The origin of the 10-a-second rhythm is still uncertain, though the evidence points to some widespread organization, probably involving the central masses as well as the cortex. There are abundant nervous connexions for coordinating the beat, and when the rhythm is well (...) developed it is possible to record impulse discharges in phase with it passing to and fro in the nerve fibers of the white matter below the cortex. But in some areas no impulse could be detected entering or leaving the cell layers, though the potential waves keep more or less in step with those elsewhere. There is no proof that the waves are kept in phase by some means which does not involve nervous signalling, for impulses in axons of small diameter might well be missed, but it is not altogether unlikely that the synchronization is due in part to a direct electrical influence of one group of nerve cells on another. Gerard and Libet have shown that synchronization can occur between the two halves of a frog's brain cut in two and then placed in contact, and more recently Arvanitaki has shown that individual nerve cells may influence one another if they are brought close together in a conducting medium. At all events there are many examples of synchronized rhythmic activity in large collections of nerve-or-muscle cells. In the optic ganglion of the water beetle, for instance, there may be both a slow rhythm when the eye is in darkness and a rapid one when the eye is exposed to a bright light. The change from a slow to a fast rhythm is curiously reminiscent of that in the cerebral cortex when a sensory stimulus abolishes the 10-a-second rhythm and substitutes one at 40–60 a second. Such resemblances in preparations of quite different structures may be quite fortuitous, but they make it difficult to resist the suggestion that the rhythms of the brain may be dependent on the general properties of cell masses rather than on any special anatomical arrangement of them. Whatever its origin, the 10-a-second rhythm of the cortex corresponds to the resting, drowsy, or inattentive state and there is no such uniform pulsation when the brain is alert.”. (shrink)

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven to (...) be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students. (shrink)

This essay discusses the instructional value of mathematical proofs using different interpretations of the analysis cum synthesis method in Apollonius’ Conics as a case study. My argument is informed by Descartes’ complaint about ancient geometers and William Thurston’s discussion on how mathematical understanding is communicated. Three historical frameworks of the analysis/synthesis distinction are used to understand the instructive function of the analysis cum synthesis method: the directional interpretation, the structuralist interpretation, and the phenomenological interpretation. I apply these interpretations to the (...) analysis cum synthesis method in order reveal how the same underlying mathematical activity occurs at different levels of scale: at the level of an individual proof, at the level of a collection of proofs, and at the level of a single line within a proof. On the basis of this investigation, I argue that the instructive value of mathematical proof lies in engendering in the reader the same mathematical activity experienced by the author themselves. (shrink)

This paper attempts to look at the contemporary mathematical practice through the lenses of the distributed cognition approach. The ubiquitous use of personal computers and the internet as a key attribute of the digital society is interpreted here as a means to achieve a more effective distribution of the human cognitive activity. The major challenge that determines the transformation of mathematical practice is identified as ‘the problem of complexity’. The computer-assisted complete formalization of mathematical proofs as a current tendency is (...) viewed as one of the strands along which the mathematical community responds to the challenge. It is shown that this tendency gives live to the project calling to revisit and rebuild the very foundations of mathematics to secure more effective communication and thus guarantee the reliability of contemporary mathematics. (shrink)

Team members co-regulate their activities and move together at the collective level of behavior while coordinating their actions toward shared goals. In parallel with team processes, team members need to resolve uncertainties arising from the changing task and environment. In this exploratory study we have measured the differential neurodynamics of seven two-person healthcare teams across time and brain regions during autonomous and collaborative segments of simulation training. The questions posed were: whether these abstract and mostly integrated constructs could be (...) separated neurodynamically; and, what could be learned about taskwork and teamwork by trying to do so? The taskwork and teamwork frameworks used were Neurodynamic Information, an electroencephalography derived measure shown to be a neurodynamic proxy for the pauses and hesitations associated with individual uncertainty, and inter-brain EEG coherence which is a required component of social interactions. No interdependency was observed between NI and IBC, and second-by-second dynamic comparisons suggested mutual exclusivity. These studies show that proxies for fundamental properties of teamwork and taskwork can be separated neurodynamically during team performances of ecologically valid tasks. The persistent expression of NI and IBC were not simultaneous suggesting that it may be difficult for team members to maintain inter-brain coherence while simultaneously reducing their individual uncertainties. Lastly, these separate dynamics occur over time frames of 15–30 s providing time for real-time detection and mitigation of individual and collaborative complications during training or live patient encounters. (shrink)

Most clinical neurofeedback studies based on functional magnetic resonance imaging use the patient's own neural activity as feedback. The objective of this study was to create a subject-independent brain state classifier as part of a real-time fMRI neurofeedback system that can guide patients with depression in achieving a healthy brain state, and then to examine subsequent clinical changes. In a first step, a brain classifier based on a support vector machine was trained from the neural information of happy autobiographical imagery (...) and motor imagery blocks received from a healthy female participant during an MRI session. In the second step, 7 right-handed female patients with mild or moderate depressive symptoms were trained to match their own neural activity with the neural activity corresponding to the “happiness emotional brain state” of the healthy participant. The training was carried out using the rt-fMRI NF system guided by the brain-state classifier we had created. Thus, the informative voxels previously obtained in the first step, using SVM classification and Effect Mapping, were used to classify the Blood-Oxygen-Level Dependent activity of the patients and converted into real-time visual feedback during the neurofeedback training runs. Improvements in the classifier accuracy toward the end of the training were observed in all the patients [Session 4–1 Median = 6.563%; Range = 4.10–27.34; Wilcoxon Test, 2-tailed p = 0.031]. Clinical improvement also was observed in a blind standardized clinical evaluation [HDRS CE2-1 Median = 7; Range 2 to 15; Wilcoxon Test, 2-tailed p = 0.016], and in self-report assessments [BDI-II CE2-1 Median = 8; Range 1–15; Wilcoxon Test, 2-tailed p = 0.031]. In addition, the clinical improvement was still present 10 days after the intervention [BDI-II CE3-2_Median = 0; Range −1 to 2; Wilcoxon Test, 2-tailed p = 0.50/ HDRS CE3-2 Median = 0; Range −1 to 2; Wilcoxon Test, 2-tailed p = 0.625]. Although the number of participants needs to be increased and a control group included to confirm these findings, the results suggest a novel option for neural modulation and clinical alleviation in depression using noninvasive stimulation technologies. (shrink)

Chapters 6 and 7 of the pseudo-Aristotelian treatise On the Cosmos display ‘a series of well-crafted and carefully organized analogies’ in order to represent the power of god pervading the whole universe. The last analogy, which is by far the most important in this section, compares the rule of god over the world to the rule of the law in a Greek city. As shown by the author in the previous analogies, the perfect order of the universe is the result (...) of the continuous creation and dissolution of single things: this process—based upon the harmony of opposites—is the keystone of the eternity and equilibrium of our world. Similarly, the law is the unmoved mover of every activity and experience in the city: both positive and negative situations involving single citizens contribute to the supreme order and stability of the city. Positive examples include the activity of rulers, officials and members of the assemblies, whereas negative examples include those who go to trial defending themselves and those who are imprisoned and destined to capital punishment. In spite of their difference, all of these actions are due to one single order, that is, the civic law, which ensures the stability of the city. To stress and illustrate this concordia discors, which characterizes both the city and the universe, the author of the treatise closes the passage with a quotation from Sophocles’ Oedipus Tyrannus :πόλις δ' ὁμοῦ μὲν θυμιαμάτων γέμει,ὁμοῦ δὲ παιάνων τε καὶ στεναγμάτωνThe author reads these verses as a perfect example of a context characterized by opposite situations: in fact, the city is full of paeans, which are interpreted as ‘songs of joy and relief’, and, at the same time, it is also full of laments and mourns. The same interpretation can be found in the Latin translation of the treatise, which gives even more emphasis to the opposition between life and death: uideasque illam ciuitatem pariter spirantem Panchaeis odoribus et graueolentibus caenis, resonantem hymnis et carminibus et canticis, eandem etiam lamentis et ploratibus heiulantem. (shrink)

In this paper, I present a novel paradox that pertains to a variety of representational states and activities. I begin by proving that there are certain contingently true propositions that no one can occurrently believe. Then, I use this to develop a further proof by which I derive a contradiction, thus giving us the paradox. Next, I differentiate the paradox from the Liar Paradox, and I show how a common response to the different variations of the Liar Paradox (...) fails to avoid the type of paradox provided in this paper. Finally, I demonstrate how the general ideas behind the paradox regarding occurrent belief can be extended to a wide range of other representational states and activities. (shrink)

LEGO and math are both about what one do with the objects. In LEGO, he/she can build sets following the instructions, or alternatively dump a whole bunch of LEGO on the floor and build whatever he/she like. In math, he/she have a similar freedom to create new things, solve problems, and play around. Geometry makes far greater use of pictures and diagrams than tends to be the case for other areas of mathematics. This chapter focuses on diagrammatic proofs as a (...) key case where proofs guide him/her through a series of actions. If one accepts the LEGO account of diagrammatic proofs then he/she has strongly sided with the geometers against Plato. The main principle of departure from Plato is to focus on mathematical activities. The same thought is meant to apply to LEGO: the particular bricks are only important insofar as they facilitate the things we can practically do with them. (shrink)

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This chapter claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its (...) claim, the chapter discusses the Kumo and Polymath projects, both of which employ Web‐based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This chapter claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the chapter discusses the Kumo and Polymath projects, both of which employ Web‐based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)