ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us (...) to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)
Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...) conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction. (shrink)
What sorts of epistemic virtues are required for effective mathematical practice? Should these be virtues of individual or collective agents? What sorts of corresponding epistemic vices might interfere with mathematical practice? How do these virtues and vices of mathematics relate to the virtue-theoretic terminology used by philosophers? We engage in these foundational questions, and explore how the richness of mathematical practices is enhanced by thinking in terms of virtues and vices, and how the philosophical picture is challenged by the complexity (...) of the case of mathematics. For example, within different social and interpersonal conditions, a trait often classified as a vice might be epistemically productive and vice versa. We illustrate that this occurs in mathematics by discussing Gerovitch’s historical study of the aggressive adversarialism of the Gelfand seminar in post-war Moscow. From this we conclude that virtue epistemologies of mathematics should avoid pre-emptive judgments about the sorts of epistemic character traits that ought to be promoted and criticised. (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)
Any paternalistic obligation a salesperson might have toward a client isnot, as was previously argued, determined or grounded in his/her being in a position of superior knowledge. Rather, the obligation stems first and most basically from the principle of non-maleficence. Beyond that, however, the particulars of any such obligation: who is vulnerable to being harmed, the harm(s) that might occur and even the kinds of actions that can reasonably be taken to protect a client from such harm, all flow from (...) the fact that the salesperson-client relationship is fundamentally one of dependency and trust.Various reasons are given to support this view and to indicate both the comprehensiveness and the fruitfulness of this way of perceiving the sales situation. In particular, the argument rules out of consideration what would, on analysis, be self-defeating or contradictory behavior on the part of the client and it helps explain why a salesperson is not obligated to certain behaviors. (shrink)
We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...) to highlight a potential danger of intellectual enculturation. (shrink)
In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. (...) I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal. (shrink)
Adult number representations can belong to either of two types. One is discrete, language-specific, and culturally-derived; the other is analog and language-independent. Quantitative evidence is presented to demonstrate that analog number representations are adult-like in young children. Five- to 7-year-olds accurately estimated rapidly presented groups of 5--11 items. Groups were presented in random order and random arrangements controlling for overall area. Children's data were qualitatively, and to some degree quantitatively, similar to adult data with one exception: the ratio of the (...) standard deviation of estimates to mean estimates decreased with age. (shrink)
Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...) meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)
It seems to be the case that when we look at a flower in the way that the scientist does, we see the flower in one way, but when we look at the flower in a way as to view it as a thing of beauty, charm, elegance, we see it in a different way; we see it as an aesthetic object. Viewing the flower in such a way as to see it, or any object, as an aesthetic object, is (...) to be in the aesthetic attitude. What this work means to do is to formulate necessary and sufficient conditions for adoption of the aesthetic attitude. ;Analytically and traditionally, the aesthetic attitude is an attitude or state-of-perceiving that is entered into by an agent which serves to make the agent receptive to the having of an aesthetic experience, and transform the object of the agent's attention from an object-in-the-world into an aesthetic object. ;The aesthetic attitude has figured prominently in aesthetics from the Enlightenment until the present. Its most important formulations are disinterestedness , Psychical Distance , Aldrich's Impressionistic Viewing, Scruton's Empiricistic Account, and the naturalistic work of John Dewey and Monroe Beardsley. I conclude that a naturalistic formulation of the aesthetic attitude is correct. (shrink)
Der Beitrag befasst sich mit den Chancen und Risiken einer allfälligen gesetzlichen Legalisierung der Suizidbeihilfe. Die Argumente, die gegen eine solche Legalisierung sprechen, werden zu drei thematischen Gruppen zusammengefasst und erörtert: „Slippery-Slope“-Argumente, Argumente vom „moralischen Druck“, und die Furcht vor einer „Entsolidarisierung der Gesellschaft“ sowie die „Gefährdung des Arzt-Patient-Verhältnisses“. Diese Gegenargumente erweisen sich als nicht zwingend, sofern Kriterien und Richtlinien für eine legitime Form der Suizidbeihilfe entwickelt und staatlich kontrolliert werden könnten.
In Art in Context: Understanding Aesthetic Value, philosopher David Fenner presents a straightforward, accessible overview of the arguments about the importance of considering the relevant context in determining the true merit of a work of ...
In this paper, I explore some parallels and dissimilarities between aesthetic appreciation that takes as its focus art objects and that which focuses on natural objects. I cover three areas. The first deals with general approach, whether a paradigm of engagement is more appropriate to environmental aesthetics than one of detachment and disinterest. The second theme is about preservation and whether the appropriate model is static or dynamic. The final theme is about environmental criticism and the application of aesthetic theory (...) to arguments for preservation. (shrink)
ZusammenfassungDer Beitrag befasst sich mit den Chancen und Risiken einer allfälligen gesetzlichen Legalisierung der Suizidbeihilfe. Die Argumente, die gegen eine solche Legalisierung sprechen, werden zu drei thematischen Gruppen zusammengefasst und erörtert: „Slippery-Slope“-Argumente, Argumente vom „moralischen Druck“, und die Furcht vor einer „Entsolidarisierung der Gesellschaft“ sowie die „Gefährdung des Arzt-Patient-Verhältnisses“. Diese Gegenargumente erweisen sich als nicht zwingend, sofern Kriterien und Richtlinien für eine legitime Form der Suizidbeihilfe entwickelt und staatlich kontrolliert werden könnten.
We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that θ does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev . We apply the result to show that Φ′ does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new (...) characterization of effectively simple sets and show that simple sets are not btt-cuppable. (shrink)
A set A is m-reducible to B if and only if there is a polynomial-time computable function f such that, for all x, x∈ A if and only if f ∈ B. Two sets are: 1-equivalent if and only if each is m-reducible to the other by one-one reductions; p-invertible equivalent if and only if each is m-reducible to the other by one-one, polynomial-time invertible reductions; and p-isomorphic if and only if there is an m-reduction from one set to the (...) other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: P = PSPACE. Every two 1-equivalent sets are p-isomorphic. Every two p-invertible equivalent sets are p-isomorphic. (shrink)
In this paper, I revisit Mattias Kumm's work on a ‘cosmopolitan conception of law’. I make two claims: First, I claim that although some criticism can be resisted by Kumm, under closer methodological scrutiny there are flaws in his theory. Second, I claim that these flaws challenge Kumm's approach when reading the Charter of the United Nations (UN Charter) as a ‘global constitution’. This also has pertinent practical implications for the functioning of the United Nations. This contribution does not take (...) a stance on the nature of law but focuses on this conception in the context of politics and law. In a first section, I recount Kumm's cosmopolitan conception of law. In a second section, I claim that implicit monism in the relationship between national and international law and theoretical idealization pose serious difficulties to the cosmopolitan approach. In a third section, I claim that these flaws pose a challenge to the United Nations when considering the UN Charter as a 'global constitution'. I sum up my findings in a final section and reflect on a future outlook for research on global constitutionalism in political philosophy. (shrink)
: In this paper, I lay out a case for why those objects of aesthetic attention which are principally characterized as natural objects should be understood not statically, as existing in merely a three-dimensional fixed state, but as dynamic, as existing in a space-time context, complete with change, movement, and flux. After this, I explain why this is important, how the dynamic nature of natural objects raises a concern for aesthetically evaluating natural objects, and how that concern may be addressed.
In this paper, I lay out a case for why those objects of aesthetic attention which are principally characterized as natural objects should be understood not statically, as existing in merely a three-dimensional fixed state, but as dynamic, as existing in a space-time context, complete with change, movement, and flux. After this, I explain why this is important, how the dynamic nature of natural objects raises a concern for aesthetically evaluating natural objects, and how that concern may be addressed.
Angesichts der unter der Führungsrolle der natur- und sozial-technologisch relevanten Wissenschaften vollzogenen extremen Spezialisierung und Ausdifferenzierung des Wissens-systems in immer engere Disziplinen mit spezialistischem Fachwissen scheint das Bemühen um ,,Weisheit", das in der Bezeichnung unserer Arbeitsbereiche steckt, gänzlich obsolet geworden zu sein. Zunächst soll die Verdrängung der Weisheitsliebe bzw. -suche anhand einiger historisch-kultureller Etappen der veränderten Bedingungen der Wissensinstitutionen und Gesellschaftsstrukturen nachgezeichnet werden . Danach wird das neuzeitliche Wissenschaftsideal mit dem antiken Weisheitsideal kontrastiert, und die Philosophie in diesem Spannungsfeld geortet (...) . Schließlich wird mit Blick auf die aktuelle Weisheitsrenaissance in verschiedenen Wissenschaften ein integrativer Ansatz eines gegenwartsfähigen Weisheitskonzeptes entworfen. (shrink)
Negative eugenics, purposive practices to eliminate some trait from our progeny, is a topic that commands discussion today. We have had the ability to practice negative eugenics for many years, perhaps for our entire history in one form or another, but today we have many options, several quite scientifically sophisticated, for such practices. What concerns me is that the easier is becomes to practice negative eugenics, the greater is the need for some consistent criterion of what makes a given trait (...) properly erasable. In this paper, I will argue that there is a need for a criterion of what makes a given trait properly a candidate for elimination through programs and practices of negative eugenics. I will begin by presenting the problem, then will proceed to explore the topic through a series of three questions: First, can we engage in programs of negative eugenics? Second, should we engage in any such programs? Third, which programs should we practice, and how should they be practiced? The paper will conclude with my own suggested criteria. (shrink)
Speculatively, it is suggested that variola virus, the cause of smallpox, evolved from an orthopoxvirus of animals of the central African rain forests (possibly now represented by Tatera poxvirus), some thousands of years ago, and first became established as a virus specific for human beings in the dense populations of the Nile valley perhaps five thousand years ago. By the end of the first millennium of the Christian era, it had spread to all the densely populated parts of the Eurasian (...) continent and along the Mediterranean fringe of north Africa. It became established in Europe during the times of the Crusades. The great voyages of European colonization carried smallpox to the Americas and to Africa south of the Sahara. Transported across the Atlantic by Europeans and their African slaves, it played a major role in the conquest of Mexico and Peru and the European settlement of north America. Variolation, an effective preventive inoculation, was devised as early as the tenth century. In 1798 this practice was supplanted by Jenner's cowpox vaccine. In 1967, when the disease was still endemic in 31 countries and caused ten to fifteen million cases and about two million deaths annually, the World Health Organization embarked on a programme that was to see the disease eradicated globally just over ten years later, and the world was formally declared to be free of smallpox in May 1980. Smallpox is unique — a specifically human disease that emerged from some animal reservoir, spread to become a worldwide, severe and almost universal affliction, and finally underwent the reverse process to emergence, namely global eradication. (shrink)
Images as a multi-faceted topic were used to teach an introductory course in Technological Studies. Nine faculty at Davidson College taught varying modules, as well as laboratory exercises. Widely differing thought patterns between the liberal arts and technologists were observed and are described herein. The syllabus is found in Appendex A.
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