Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Penelope Deutscher reconsiders the role of procreation in Foucault's thought, especially its proximity to risk, mortality, and death. Foucault's Futures brings together his work on sexuality and biopolitics to provide new insights into the conflicted political status of reproductive conduct and what it means for feminism and critical theory.

The goal of this paper is to sketch a distinctive version of naturalism in the philosophy of science, both by tracing historical antecedents and by addressing contemporary objections.

En estos tres artículos, que aparecen por primera vez en castellano, bosqueja el autor junto a una teoría general de la representación política, una relectura del problema permanente de la clasificación de los regímenes. Destaca en ellos particularmente el tratamiento de la cuestión de las causas de la corrupción de todas las formas de gobierno, incluida la democracia. Maquiaveliano riguroso y consciente, Freund no esquiva ninguno de los grandes problemas que siguen dividiendo hoy a quienes se acercan -no siempre (...) libres de prejuicios- a la realidad tantas veces poco complaciente de la política. (shrink)

A novel treatment of an issue central to much current work in metaphysics: the distinction between the essential and accidental properties of individuals. Mackie challenges widely held views, and arrives at what she calls "minimalist essentialism," an unorthodox theory according to which ordinary individuals have relatively few interesting essential properties. Mackie's clear and accessible discussions of issues surrounding necessity and essentialism mean that the book will appeal as much to graduate students as it will to seasoned metaphysicians.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical (...) realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

We adopt and extend the concept of ‘noncooperative space’ to analyze how (aspirant) black women intellectual activists attempt to sustain their efforts within settings that publicly endorse racial equality, while, in practice, the contexts remain deeply racist. Noncooperative spaces reflect institutional, organizational, and social environments portrayed by powerful white agents as conducive to anti-racism work and promoting racial equality but, indeed, constrain individuals who challenge racism. Our work, which is grounded in intersectionality, draws on an autoethnographic account of racially motivated (...) domestic violence suffered by our lead author. Our analysis suggests that (aspirant) black women intellectual activists must develop courage to sustain their ‘voice’ within noncooperative spaces. However, the three interlinked dimensions of noncooperative spaces—namely, deceiving design, hegemonic actors’ indifference to racism, and (some assimilated gatekeepers’) false equivalence—may gradually erode a black female scholar’s courage. This forces her ‘voice’ to vanish temporarily, or even permanently. Courage is thus fragile and depletable. Yet, courage can be regenerated, resulting in regaining voice. Consequently, we propose courageous collective action by white allies and black and brown individuals who voluntarily and officially cooperate within and across various spaces to achieve racial equality. (shrink)

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions are characterized. Most of these logics are intensional in nature and possess, (...) in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches. (shrink)

A plea for natural philosophy --On the question of realism --Hume and Reid --Moore's hands --Wittgenstein on hinges --A note on truth and reference --The philosophy of logic --A Second Philosophy of logic --Psychology and the a priori sciences --Do numbers exist? --Enhanced if-thenism.

The Logical Must is an examination of Wittgenstein's philosophy of logic, early and late, from an austere naturalistic perspective called "Second Philosophy.".

In lieu of an abstract, here is a brief excerpt of the content:“Foucault for Psychoanalysis”Monique David-Ménard’s Kind of BluePenelope DeutscherFoucault for psychoanalysis? This is a paradoxical question. Foucault also produced a critique of psychoanalysis, aiming to show that sexuality was not an a-temporal reality, nor a truth eventually discovered by Freud. It was a discursive formation, one among others.—Eloge des hasards dans la vie sexuelle, 172.To the philosophers..A practicing psychoanalyst and a professor of philosophy, Monique David-Ménard extends a singular proposition (...) to philosophers. She is not the first philosopher to engage the assumptions of psychoanalysis, but hers is a particularly idiosyncratic suggestion for how to think with the analyst’s clinical work, intimate and alien. She has proposed we do so in the company of Gilles Deleuze,1 and most recently, in Eloge des hasards dans la vie sexuelle (2011) has extended her invitation to think with the clinic through an ingenious reading of Michel Foucault.As Eloge opens, David-Ménard’s reading of Foucault is far from the scene. Early in the work, we are asked to think with a practice, seemingly in a different register. An analyst finds herself accused of being an obstacle, of opposing the analysand’s wishes in life. In this context, we are asked to think about the blue of a painting in the room where the analysand waits (David-Ménard 2011, 34). [End Page 111]Given the criticisms of psychoanalysis elaborated in the work of Foucault and Deleuze, one might have expected these philosophers to be considered in Eloge in that stead.2 Or, one could imagine Foucault playing a role because of his genealogical analysis of the formation of the Oedipus complex.3Two years after the publication of Deleuze’s Anti-Oedipus, Foucault had, in Abnormal, characterized the desiring space of the family as stimulated in a capillary network of individualizing normalization (Foucault 2003, 257), linked by medicalized and expert concern of which parents became the auxiliaries. They are charged, and charge themselves, with the goals of family health, of preservative and defensive good parenting, and with the conduct of the child’s conduct. Foucault describes the pre-twentieth century emergence of lay and expert preoccupation with the dangers of childhood masturbation. This produces an environment whose intensive scrutiny involves a close “folding of... parents’ bodies over... children’s bodies” (ibid., 265) in a saturated, hermetic environment stimulating oedipalized desire. This means one can offer a genealogical account of the conditions of formation of the latter. These include a responsibilization of parents, who control for the presence of abnormal sex, advised by new experts (lay, medical, pseudo-scientific) in collective crusades for health and life. Thus there is differentiation through minor gradations of abnormality. Even the specificity of the architectural spaces in which scrutiny is directed plays a role in the hollowing out of the child’s putatively interior psychic-sexual life, the intense object of “palpating” interest (Foucault 1978, 45) in new apparatuses of knowledge-power (Foucault 2003, 327).Because he presents this as the context in which the “nuclear and substantial family was gradually constituted” (Foucault 2003, 264), along with its oedipal configurations of desire, Foucault emphasizes the contingency of both. Psychoanalysis would be, from his perspective, willfully indifferent to these or other possibilities for genealogical analysis of the apparatus of power, knowledge, and the body, of the repeating configurations of desire that are its special domain of expertise,4 and the emergent forms of authority associated with the analyst, described in History of Madness (Foucault 2009) and in Psychiatric Power (Foucault 2006).Perhaps we’d expect to see David-Ménard, the psychoanalyst, addressing these accounts, but they are neither repudiated in Eloge, nor extensively discussed. Far from rejecting a genealogical analysis of psychoanalysis, she promotes its promise. But this is so in a particular sense.Necessity and repetitionA particular patient might fear being “mad like her mother.” She might fear being denied, trapped, or annihilated (David-Ménard 2011, 33). She might [End Page 112] consider herself somehow denied a lover. She might seem to be denying herself a lover. There’s a degree of necessity in all these variants. Perhaps these... (shrink)

Sexual difference as a basis of equality : an introduction to Irigarayan politics -- Irigaray on language : from the speech of dementia to the problem of sexual indifference -- Rethinking the politics of recognition : the declaration of Irigarayan sexuate rights -- Irigarayan performativity : is this a question of can saying it make it so? -- Sexuate genre : ethics and politics for improper selves -- Anticipating sexual difference : mediation, love, and divinity -- Interrogating an unasked question (...) : is there sexual difference? -- The impossible friend : traversing the heterosocial, the homosocial, and the successes of failure -- Sexed discourse and the language of the philosophers : to be two -- Effacement redoubled? : between the Orient and the Occident. (shrink)

The work of the distinguished philosopher Sarah Kofman has, since her tragic death in 1994, become a focus for many scholars interested in contemporary French philosophy. The first critical collection on her thought to appear in English, Enigmas evaluates Kofman's most important contributions to philosophy, psychoanalytic theory, feminism, and literary theory. These insightful essays range from analyses of Kofman's first book, L'Enfance de l'art, to her last, L'Imposture de la beauté. This unique volume represents the major themes in Kofman's scholarship: (...) literature and aesthetics; philosophy and metaphor; women, feminism, and psychoanalysis; and Jews and German nationalism. Selected essays explore and diagnose Kofman's personal struggles as they are reflected in her writing. (shrink)

It’s often said that set theory provides a foundation for classical mathematics because every classical mathematical object can be modeled as a set and every classical mathematical theorem can be proved from the axioms of set theory. This is obviously a remarkable mathematical fact, but it isn’t obvious what makes it ‘foundational’. This paper begins with a taxonomy of the jobs set theory does that might reasonably be regarded as foundational. It then moves on to category-theoretic and univalent foundations, exploring (...) to what extent they do these same jobs, and to what extent they might do other jobs also reasonably regarded as foundational. (shrink)

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in (...) social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there. In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms,1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, in Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception cannot give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we do not actually acquire mathematical knowledge at all; rather, we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either – also Truth, Beauty, Justice, the Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it will not begin to satisfy a contemporary, scientifically minded philosopher.. (shrink)

What Do Philosophers Do? takes up the leading arguments for radical skepticism from an everyday point of view. A range of philosophical methods are examined and employed, for a revealing portrait of what philosophers do, and perhaps a quiet suggestion for what they should do, for what they do best.

"À son décès, Julien Freund laissait parmi ses papiers un recueil de textes inédits, sous forme épistolaire, adressés à ses amis et collègues : les Lettres de la vallée. Dans ces méditations, rédigées en sa retraite vosgienne - qui n'est pas sans évoquer la Hütte de Heidegger, de l'autre côté du Rhin - et loin des servitudes propres aux écrits universitaires, l'auteur de L'Essence du politique s'exprime en toute liberté, à bâtons rompus, sur ses sujets de prédilection (le politique, (...) l'idéologie, la guerre, les totalitarismes, l'utopie, la décadence), mais également l'enseignement, le journalisme, la connaissance, le droit, la morale, la métaphysique, les limites... Ce livre forme la clef de voûte où se rejoignent les grands thèmes de sa pensée."--Page 4 de la couverture. (shrink)

The human microbiome has become one of the dominant biomedical frameworks of the contemporary moment that may be understood to be post-Pasteurian. The recognitions the human microbiome opens up for thinking about the biological self and the individual have ontological and epistemological ramifications for considering what and who the human being is. As this article illustrates, the microbiopolitics of the human microbiome challenges the immunitarian Pasteurian model in which the organismic self shores itself up and defends itself against a microbial (...) non-self or other. Instead, this theory presents the human organism as comprised of multiple ecosystems and as a multitude, suggesting that the thanatopolitical attempts to wipe out microbial others (evident in the overuse and misuse of antibiotics, for example) are giving way to an affirmative microbiopolitics grounded in generative multispecies relationality. This article sets out to make the case for this affirmative microbiopolitics. (shrink)

Professor Deutscher studies Beauvoir's philosophy on "otherness" not just through her famous views on gender (in her celebrated 1949 work The...

I discuss a thesis that I call ‘The Appearance of Mind-Independence’, to the effect that, to the subject of an ordinary perceptual experience, it seems that the experience involves the awareness of a mind-independent world. Although this thesis appears to be very widely accepted, I argue that it is open to serious challenge. Whether such a challenge can be maintained is especially relevant to the assessment of any theory, such as Berkeley’s idealism, according to which the only objects of which (...) we are aware in perception are mind-dependent. But the issue is of significance for the philosophy of perception more generally. In the course of my sceptical discussion, I argue that recent work by Campbell and Cassam may be flawed by a failure to take sufficiently seriously the requirements for genuine mind-independence. (shrink)

We shall distinguish between sortal predication and standard predication. The former kind of predication necessarily involves sortal concepts but the latter, as it is customarily viewed, does not. It is generally thought that the only essential occurrence of a concept in a standard predication is the concept being predicated. In this paper, we shall put forward an alternative view. We shall propose to understand standard predication as a cognitive act essentially requiring sortal concepts. We shall call this view conceptual predication (...) sortalism and ground it on the basis of epistemic-semantic reasons. Concepts are understood as intersubjectively realizable capacities or abilities that fulfill particular cognitive roles, such as those of classification, categorization, individuation, and referring. (shrink)

Michael Rathjen and the present author have shown that ‐bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in. In this note we show that the base theory can be weakened to. Our argument makes crucial use of a normal function f with and. We shall also exhibit a normal function g with and.

In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is ‐definable relative to the previous stages of the recursion. It is known that this principle is provable in. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still (...) provable in. (shrink)

My aim in this paper is to propose what seems to me a distinctive approach to set theoretic methodology. By ‘methodology’ I mean the study of the actual methods used by practitioners, the study of how these methods might be justified or reformed or extended. So, for example, when the intuitionist's philosophical analysis recommends a wholesale revision of the methods of proof used in classical mathematics, this is a piece of reformist methodology. In contrast with the intuitionist, I will focus (...) more narrowly on the methods of contemporary set theory, and, more importantly, I will certainly recommend no sweeping reforms. Rather, I begin from the assumption that the methodologist's job is to account for set theory as it is practiced, not as some philosophy would have it be. This credo lies at the very heart of the so-called ‘naturalism’ to be described here.A philosopher looking at set theoretic practice from the outside, so to speak, might notice any number of interesting methodological questions, beginning with the intuitionist's ‘why use classical logic?’, but this sort of question is not a live issue for most practicing set theorists. One central question on which the philosopher's and the practitioner's interests converge is this: what is the status of independent statements like the continuum hypothesis (CH)? A number of large questions arise in its wake: what criteria should guide the search for new axioms? For that matter, what reasons support our adoption of the old axioms? (shrink)

We have previously established that [Formula: see text]-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over [Formula: see text]. In this paper, we show that the base theory can be lowered to [Formula: see text]. We also show that the minimal Bachmann–Howard fixed point of a dilator [Formula: see text] can be represented by a notation system [Formula: see text], which is computable relative to [Formula: see text]. The statement that [Formula: see text] (...) is well founded for any dilator [Formula: see text] will still be equivalent to [Formula: see text]-comprehension. Thus, the latter is split into the computable transformation [Formula: see text] and a statement about the preservation of well-foundedness, over a system of computable mathematics. (shrink)