StillwellJohn.* * _ A Concise History of Mathematics for Philosophers. _ Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2019. Pp. 69. ISBN: 978-1-108-45623-4, 978-1-108-61012-4. doi.org/10.1017/9781108610124.

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of intuition (...) in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

In this paper I argue that Kant's distinction in the Inaugural Dissertation between the sensible and the intelligible arises in part out of certain open questions left open by his comparison between mathematics and metaphysics in the Prize Essay. This distinction provides a philosophical justification for his distinction between the respective methods of mathematics and metaphysics and his claim that mathematics admits of a greater degree of certainty. More generally, this illustrates the importance of Kant's reflections on mathematics for the (...) development of his Critical philosophy. (shrink)

In her recent book, Realism in mathematics, Penelope Maddy attempts to reconcile a naturalistic epistemology with realism about set theory. The key to this reconciliation is an analogy between mathematics and the physical sciences based on the claim that we perceive the objects of set theory. In this paper I try to show that neither this claim nor the analogy can be sustained. But even if the claim that we perceive some sets is granted, I argue that Maddy's account fails (...) to explain the key issue faced by an epistemology for mathematics, namely the step from knowledge of the finite to knowledge of the infinite. (shrink)

By way of these investigations, we hope to understand better the rationale behind Kant's theory of intuition, as well as to grasp many facets of the relations ...

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...) mathematics and transcendental philosophy. (shrink)

This thesis attempts to argue against an influential interpretation of Kant's philosophy of mathematics according to which the role of pure intuition is primarily logical. Kant's appeal to pure intuition, and consequently his belief in the synthetic character of mathematics, is, on this view, a result of the limitations of the logical resources available in his time. In contrast to this, a reading is presented of the development of Kant's philosophy of mathematics which emphasises a much richer philosophical role for (...) intuition, beyond that of filling in deductive gaps in mathematical arguments. This role is largely determined by two factors: on the one hand, the epistemological gap in a traditional account of mathematical certainty, and on the other hand, the metaphysical worry raised by the followers of Leibniz and Wolff regarding the status of mathematics as a science consisting of truths. ;By appreciating how Kant's philosophy of mathematics developed against the background of what he saw as a conflict between the 'metaphysicians' and the 'mathematicians', we can see how the doctrine of pure intuition was required to fill the gaps in the account of the method and certainty of mathematics which Kant presents in the Prize Essay of 1764. This conflict takes its sharpest form between the monadists who uphold the metaphysical necessity of simple elements, and the geometers who uphold the infinite divisibility of space. In the Prize Essay Kant indirectly addresses this conflict by comparing the two disciplines. He asserts that the method of attaining certainty in mathematics is different from the method of metaphysics, and that as a result of this difference--primarily the different roles of definitions in each--mathematics is capable of a higher degree of certainty than metaphysics. He does not, however, explain this difference. Such an explanation seems particularly important in light of Kant's wish to secure geometry against the challenge of metaphysics. This thesis attempts to show that it is to provide such an explanation, to provide a philosophical grounding for the Prize Essay account, that Kant invokes the doctrine of pure intuition in his later account of mathematical knowledge. (shrink)

The broadly-stated aim of this rich collection is to reevaluate and reconceptualize the mathematization thesis, which the editors take to signify “above all the transformation of scientific concepts and methods, especially those concerning the nature of matter, space, and time, through the introduction of mathematical techniques and ideas”. As a historiographical thesis, it is the thesis that “the scientific revolution, and by implication modern science as a whole, is guided by the project of mathematization”.In the introduction to the volume, the (...) editors acknowledge the virtues of the historiographical thesis, which explain its persistence. For example, it highlights a constitutive feature... (shrink)

This is a book of wide-ranging scope, as the title suggests. First, it canvasses a broad selection of topics—from electromagnetism and quantum mechanics to Husserl’s phenomological constitution of logic, from Russell and Wittgenstein to Hartry Field. Second, its aims are broad. The author describes the book both as a “rational reconstruction of Poincaré’s position” and as a “treatise on modern epistemology”. The former description is somewhat misleading in that, together with Zahar’s stated aim of both “clarifying and of then reconciling (...) Poincaré’s various theses about the foundations of mathematics and the natural sciences”, it might give the reader hope that all of these theses might be shown to be part of one unified position. In fact, though, Poincaré’s philosophy of physics and his philosophy of mathematics turn out to be fundamentally different. Zahar attributes to Poincaré a “structural realist” view of physics and a “quasi-Kantian” constructivist view of mathematics. It’s not entirely clear, though, why the considerations adduced in favor of structural realism in physics can’t be carried over to the case of mathematics. More might have been said to explain this divergence, perhaps for example by developing the contrast in the introduction between the empirical testability of scientific hypotheses and the intuitive self-evidence of mathematical theorems. (shrink)

This book is a welcome contribution to the literature on Kant's philosophy of mathematics in two particular respects. First, the author systematically traces the development of Kant's thought on mathematics from the very early pre-Critical writings through to the Critical philosophy. Secondly, it puts forward a challenge to contemporary Anglo-Saxon commentators on Kant's philosophy of mathematics which merits consideration.A central theme of the book is that an adequate understanding of Kant's pronouncements on mathematics must begin with the recognition that mathematics (...) in Kant's time was poised at the beginning of what Pierobon calls the ‘algebraic revolution’ of the nineteenth century. For Kant, Euclidean geometry, with its heavy reliance on the geometric image, was the paradigm of certainty. The algebraic revolution of the nineteenth century replaced that paradigm with an algebraic formalism, thereby freeing mathematics from any connection to the geometric image, and also severing the link to intuition. Pierobon describes this as the ‘divergence between the image and writing [l'écriture]’. So great was the shift, Pierobon suggests, that, after the developments of the nineteenth century, it became difficult to find any sense in Kant's conception of mathematics as sensible knowledge. This, certainly, was the view of Russell, who notoriously claimed in Mysticism and Logic that modern developments in logic dealt a ‘fatal blow to the Kantian philosophy’ and that ‘the whole doctrine of a priori intuitions, by which Kant explained the possibility of pure mathematics, is wholly inapplicable to mathematics in its present form’.1 Pierobon claims, though, that much of Anglo-Saxon commentary on Kant's philosophy of mathematics begins from this ‘rationalist and logicist’ position, reading Kant's philosophy of mathematics from a post-algebraic-revolution perspective. This book attempts to offer a corrective to that position by offering a Kantian conception of mathematics …. (shrink)

I begin by discussing views about the permissibility of lying by political leaders. Sections II and III address historically important lies and lies about history and the historical record. These two categories overlap - some lies about the historical record were historically important events. In section IV, I discuss the related notion of half-truths and give examples of misleading/deceptive half-truths about history. In the final section of this chapter, I briefly discuss the obligations of historians to give truthful accounts of (...) historical events. (shrink)

An incisive exploration of the way Greek myths empower us to defeat tyranny. As tyrannical passions increasingly plague twenty-first-century politics, tales told in ancient Greek epics and tragedies provide a vital antidote. Democracy as a concept did not exist until the Greeks coined the term and tried the experiment, but the idea can be traced to stories that the ancient Greeks told and retold. From the eighth through the fifth centuries BCE, Homeric epics and Athenian tragedies exposed the tyrannical potential (...) of individuals and groups large and small. These stories identified abuses of power as self-defeating. They initiated and fostered a movement away from despotism and toward broader forms of political participation. Following her highly praised book Enraged: Why Violent Times Need Ancient Greek Myths, the classicist Emily Katz Anhalt retells tales from key ancient Greek texts and proceeds to interpret the important message they hold for us today. As she reveals, Homer's Iliad and Odyssey, Aeschylus's Oresteia, and Sophocles's Antigone encourage us—as they encouraged the ancient Greeks—to take responsibility for our own choices and their consequences. These stories emphasize the responsibilities that come with power (any power, whether derived from birth, wealth, personal talents, or numerical advantage), reminding us that the powerful and the powerless alike have obligations to each other. They assist us in restraining destructive passions and balancing tribal allegiances with civic responsibilities. They empower us to resist the tyrannical impulses not only of others but also in ourselves. In an era of political polarization, Embattled demonstrates that if we seek to eradicate tyranny in all its toxic forms, ancient Greek epics and tragedies can point the way. (shrink)

About half of the undergraduate and roughly 40 percent of graduate degree recipients in science and engineering are women. As increasing numbers of these women pursue research careers in science, many who choose to have children discover the unique difficulties of balancing a professional life in these highly competitive (and often male-dominated) fields with the demands of motherhood. Although this issue directly affects the career advancement of women scientists, it is rarely discussed as a professional concern, leaving individuals to face (...) the dilemma on their own. To address this obvious but unacknowledged crisis—the elephant in the laboratory, according to one scientist—Emily Monosson, an independent toxicologist, has brought together 34 women scientists from overlapping generations and several fields of research—including physics, chemistry, geography, paleontology, and ecology, among others—to share their experiences. From women who began their careers in the 1970s and brought their newborns to work, breastfeeding them under ponchos, to graduate students today, the authors of the candid essays written for this groundbreaking volume reveal a range of career choices: the authors work part-time and full-time; they opt out and then opt back in; they become entrepreneurs and job share; they teach high school and have achieved tenure. The personal stories that comprise Motherhood, the Elephant in the Laboratory not only show the many ways in which women can successfully combine motherhood and a career in science but also address and redefine what it means to be a successful scientist. These valuable narratives encourage institutions of higher education and scientific research to accommodate the needs of scientists who decide to have children. (shrink)

How should we distinguish between ethical and unethical ways of pursuing profit in a market? The market failures approach (MFA) to business ethics purports to provide an answer to this question. I argue that it fails to do so. The source of this failure is the MFA’s reliance on Pareto efficiency as a core ethical principle. Many ethically “preferred” tactics for seeking profit cannot be justified by appeal to Pareto efficiency. One important reason for this is that Pareto efficiency, as (...) understood by the theory of welfare economics upon which the MFA relies, assumes a static conception of efficiency. This is a problem because many ethically “preferred” tactics can only be justified by appeal to dynamic efficiency considerations. I argue that, instead of Pareto efficiency, we should look to the value of wealth creation to understand the ethical constraints on how market actors may pursue profit. (shrink)

Play is a vital component of the social life and well-being of both children and adults. This book examines the concept of play and considers a variety of the related philosophical issues. It also includes meta-analyses from a range of philosophers and theorists, as well as an exploration of some key applied ethical considerations. The main objective of The Philosophy of Play is to provide a richer understanding of the concept and nature of play and its relation to human life (...) and values, and to build disciplinary and paradigmatic bridges between scholars of philosophy and scholars of play. Including specific chapters dedicated to children and play, and exploring the work of key thinkers such as Plato, Sartre, Wittgenstein, Gadamer, Deleuze and Nietzsche, this book is invaluable reading for any advanced student, researcher or practitioner with an interest in education, playwork, leisure studies, applied ethics or the philosophy of sport. (shrink)

Many business ethicists assume that if a type of conduct is illegal, then it is also unethical. This article scrutinizes that assumption, using the rideshare company Uber’s illegal operation in the city of Philadelphia as a case study. I argue that Uber’s unlawful conduct was permissible. I also argue that this position is not an extreme one: it is consistent with a variety of theoretical commitments in the analytic philosophical tradition regarding political obligation. I conclude by showing why business ethicists (...) would have a better rejoinder to the “dominant view” of business ethics associated with Milton Friedman if they dispensed with the assumption that illegal implies unethical. (shrink)

Interpreters are less univocal than one might think in assessing Søren Kierkegaard's attitude toward eudaimonism. Through an analysis of several key texts from across Kierkegaard's authorship, I argue that existing interpretations do not convincingly address the relationship between Kierkegaard's critique of eudaimonism and his mid-nineteenth-century context, which was dominated by post-Kantian idealists. While I am sympathetic to aspects of deontological and aretaic interpretations, a contextual reading shows that his critique centers on what he diagnoses as the enclosure of the modern (...) self. This puts his critique of eudaimonism in the purview of his moral psychology and in continuity with his critique of romanticism. (shrink)

_Philosophers consider race and racism from the perspective of lived, bodily experience._.

Research on the specific features of daydreaming compared with mind-wandering and night dreaming is a neglected topic in the philosophy of mind and the cognitive neuroscience of spontaneous thought. The extant research either conflates daydreaming with mind-wandering (whether understood as task-unrelated thought, unguided attention, or disunified thought), characterizes daydreaming as opposed to mind-wandering (Dorsch, 2015), or takes daydreaming to encompass any and all “imagined events” (Newby-Clark & Thavendran, 2018). These dueling definitions obstruct future research on spontaneous thought, and are insufficiently (...) precise to guide empirical studies. They also fail to illuminate the phenomenal core of daydreaming, namely, its dreamlike qualities. Although daydreaming is related to both mind-wandering and narrative imagination, it is not reducible to either. We argue that daydreams are experiences of spontaneous, immersive imagination in the waking state. The main task of our investigation is to distinguish daydreaming, conceptually and phenomenologically, from mind-wandering, on the one hand, and night dreaming, on the other. Although daydream experiences can vary widely, we distinguish prototypical experiences of daydreaming from adjacent imaginative activity, including fleeting imagery and “focused daydreaming,” or crafted visualization. We consider our phenomenological analysis as preparatory work for conceptually distinguishing different spontaneous and imaginative states so that they can be investigated accordingly with questionnaires and qualitative methods. We argue that precision about the phenomenal character of daydreaming can guide neurophenomenological investigations, help delimit studies on individual variance in daydreaming features, and identify differences among daydreaming, mind-wandering, and night dreaming conceptually and phenomenologically, and possibly eventually in terms of neural correlates. (shrink)

Buying local is a prominent form of ethical consumption. We commonly assume that products that are local are in some respect ethically superior to ones that are not. This article contributes to research on local food by scrutinizing this assumption in light of some central values of the locavore movement. It identifies four central ethical causes from prior literature on locavorism: protecting the environment, promoting community, promoting small business, and contributing to the prosperity of one’s local economy. It then analyzes (...) whether the contribution of buying local to these causes can justify the general perception that buying local is a good way to be an ethical consumer. Its main finding is that these justifications fail to provide a strong positive ethical reason for consumers in general to adopt the practice of buying local. (shrink)

Thomas Carson offers the most comprehensive and up-to-date investigation of moral and conceptual questions about lying and deception. Part I addresses conceptual questions and offers definitions of lying, deception, and related concepts such as withholding information, "keeping someone in the dark," and "half truths." Part II deals with questions in ethical theory. Carson argues that standard debates about lying and deception between act-utilitarians and their critics are inconclusive because they rest on appeals to disputed moral intuitions. He defends (...) a version of the golden rule and a theory of moral reasoning. His theory implies that there is a moral presumption against lying and deception that causes harm — a presumption at least as strong as that endorsed by act-utilitarianism. He uses this theory to justify his claims about the issues he addresses in Part III: deception and withholding information in sales, deception in advertising, bluffing in negotiations, the duties of professionals to inform clients, lying and deception by leaders as a pretext for fighting wars, and lying and deception about history, and cases of distorting the historical record by telling half-truths. The book concludes with a qualified defence of the view that honesty is a virtue.Readership: Scholars and advanced students of philosophy, as well as historians and students of history. (shrink)

Emily Elizabeth Constance Jones was an English logician and contemporary of Bertrand Russell, as well as Mistress of Girton College, Cambridge. In this book, originally published in 1911, she argues for the existence of another fundamental law of thought to join the Law of Contradiction and the Law of Excluded Middle: the Law of Significant Assertion. This book will be of value to anyone with an interest in logic or in Jones' work.

Incomplete Archaeologies takes a familiar archaeological concept--assemblages--and reconsiders such groupings, collections and sets of things from the perspective of the work required to assemble them. The discussions presented here engage with the practices of collection, construction, performance and creation in the past (and present) which constitute the things and groups of things studied by archaeologists--and examine as well how these things and thing-groups are dismantled, rearranged, and even destroyed, only to be rebuilt and recreated. The ultimate aim is to reassert (...) an awareness of the incompleteness of assemblage, and thus the importance of practices of assembling (whether they seem at first creative or destructive) for understanding social life in the past as well as the present. The individual chapters represent critical engagements with this aim by archaeologists presenting a broad scope of case studies from Eurasia and the Mediterranean. Case studies include discussions of mortuary practice from numerous angles, the sociopolitics of metallurgy, human-animal relationships, landscape and memory, the assembly of political subjectivity and the curation of sovereignty. These studies emphasise the incomplete and ongoing nature of social action in the past, and stress the critical significance of a deeper understanding of formation processes as well as contextual archaeologies to practices of archaeology, museology, art history, and other related disciplines. Contributors challenge archaeologists and others to think past the objects in the assemblage to the practices of assembling, enabling us to consider not only plural modes of interacting with and perceiving things, spaces, human bodies and temporalities in the past, but also to perhaps discover alternate modes of framing these interactions and relationships in our analyses. Ultimately then, Incomplete Archaeologies takes aim at the perceived totality not only of assemblages of artefacts on shelves and desks, but also that of some of archaeology's seeming-seamless epistemological objects"--From publisher's website. (shrink)