S. SHAPIRO (ed.), Intensional Mathematics (Studies in Logic and the Foundations of Mathematics, vol. 11 3). Amsterdam: North-Holland, 1985. v + 230 pp. $38.50/100Df.

Review of John Burgess' and Gideon Rosen's A Subject with no Object.

An Essay on Mathematical Instrumentalism M. Detlefsen. THE PHILOSOPHICAL FUNDAMENTALS OF HILBERT'S PROGRAM 1. INTRODUCTION In this chapter I shall attempt to set out Hilbert's Program in a way that is more revealing than ...

In this paper I have considered various attempts to attribute significance to Gödel's second incompleteness theorem (G2 for short). Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are false. Two others (an argument suggested by Beth, Cohen and ??? and Resnik's Interpretation), I argue, are groundless.

Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I consider possible arguments for this view. Major elements of this argument are out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They also require a special form of consistency requirement.

Introduction to a special issue of Philosophia Mathematica on the mechanization of reasoning. Authors include: M. Detlefsen, D. Mundici, S. Shanker, S. Shapiro, W. Sieg and C. Wright.

SOME suggestions are here made for improvement of the text and understanding of Pliny's Eastern Aegean geography. The editions studied for the purpose are Detlefsen's special edition of the geographical books and Mayhoff's Teubner vol. i . The citations of MSS. readings given below are normally taken from Mayhoff's apparatus, which gives a fuller coverage than Detlefsen's. The MSS. are cited by the letters given them in Mayhoff's edition and the Budé Pliny book i , pp. 37 f. (...) One further MS., hitherto not collated in the geographical books, is here cited. Following Campbell, who first drew attention to it, I cite it as c. This MS., formerly in the Phillipps collection in Cheltenham and now in the Pierpont Morgan Library , is assigned to Lorsch and said to be of the first half of the ninth century. It seems thus to be somewhat older than A and E, and in that case is the oldest known surviving MS. of the geographical books. (shrink)

SOME suggestions are here made for improvement of the text and understanding of Pliny's Eastern Aegean geography. The editions studied for the purpose are Detlefsen's special edition of the geographical books and Mayhoff's Teubner vol. i. The citations of MSS. readings given below are normally taken from Mayhoff's apparatus, which gives a fuller coverage than Detlefsen's. The MSS. are cited by the letters given them in Mayhoff's edition and the Budé Pliny book i, pp. 37 f. One further (...) MS., hitherto not collated in the geographical books, is here cited. Following Campbell, who first drew attention to it, I cite it as c. This MS., formerly in the Phillipps collection in Cheltenham and now in the Pierpont Morgan Library, is assigned to Lorsch and said to be of the first half of the ninth century. It seems thus to be somewhat older than A and E, and in that case is the oldest known surviving MS. of the geographical books. (shrink)

An outstanding reference source for the wide range of philosophical contributions made by women writing in Europe from about 1560 to 1780. It shows the range of genres and methods used by women writing in these centuries in Europe, thus encouraging an expanded understanding of our historical canon.

The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.

© The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] author makes clear that he does not see this book as a contribution to the philosophy of mathematics as traditionally understood. He takes it instead to be an essay about the philosophy of mathematics, one whose purpose is to explain its existence and to make clear the limited extent to which its current and past forms are properly regarded as philosophies of mathematics per (...) se.A little more exactly, the aim is to explain the seemingly ‘perennial’ interest1 western philosophers have had in mathematics 2 and how this became significant enough to give rise to and sustain an identified sub-discipline of philosophy....Hacking identifies two principal reasons for this, both of which he sees as having originated in what might broadly be termed our experience of mathematics. The first of... (shrink)

In 2020, the University of Pennsylvania instituted a graduate certificate in public philosophy. In many ways, this certificate formalized and recognized the public engagement work that graduate students in the philosophy department and beyond had been involved with for some years. One element of the certificate, however, was pivotal in moving our work in public philosophy forward in important ways. This element is the research seminar in public philosophy. In this paper, we recount the motivation for the creation of the (...) certificate and especially the motivation for the inclusion of the research seminar. We also explore ways in which such a certificate, along with the deliberately self-reflective work of the research seminar, might help us reimagine the nature and value of philosophy and its connection with human life and flourishing. We focus on metaphilosophical themes such as the very nature of philosophy and the philosopher as well as the importance of cultivating a new generation of academic philosophers committed to transcending the distinction between the academy and the public and, relatedly, between academic philosophy and public philosophy. (shrink)

There have been many different historical-intellectual accounts of the shaping and development of concepts of liberty in pre-Enlightenment Europe. This volume is unique for addressing the subject of liberty principally as it is discussed in the writings of women philosophers, and as it is theorized with respect to women and their lives, during this period. The volume covers ethical, political, metaphysical, and religious notions of liberty, with some chapters discussing women's ideas about the metaphysics of free will, and others examining (...) the topic of women's freedom in their moral and personal lives as well as in the public socio-political domain. In some cases, these topics are situated in relation to the emergence of the concept of autonomy in the late eighteenth century, and in others, with respect to recent feminist theorizing about relational autonomy and internalized oppression. Many of the chapters draw upon a wide range of genres, including polemical texts, poetry, plays, and other forms of fiction, as well as standard philosophical treatises. Taken as a whole, this volume shows how crucial it is to recover the too-long forgotten views of female and women-friendly male philosophers of the seventeenth and eighteenth centuries. In the process of recovering these voices, our understanding of philosophy in the early modern period is not only expanded, but also significantly enhanced, toward a more accurate and gender-inclusive history of our discipline. (shrink)

According to Margaret Cavendish the entire natural world is essentially rational such that everything thinks in some way or another. In this paper, I examine why Cavendish would believe that the natural world is ubiquitously rational, arguing against the usual account, which holds that she does so in order to account for the orderly production of very complex phenomena (e.g. living beings) given the limits of the mechanical philosophy. Rather, I argue, she attributes ubiquitous rationality to the natural world in (...) order to ground a theory of the ubiquitous freedom of nature, which in turn accounts for both the world's orderly and disorderly behavior. (shrink)

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...) discuss its epistemological significance. (shrink)

Between 1653 and 1655 Margaret Cavendish makes a radical transition in her theory of matter, rejecting her earlier atomism in favour of an infinitely-extended and infinitely-divisible material plenum, with matter being ubiquitously self-moving, sensing, and rational. It is unclear, however, if Cavendish can actually dispense of atomism. One of her arguments against atomism, for example, depends upon the created world being harmonious and orderly, a premise Cavendish herself repeatedly undermines by noting nature’s many disorders. I argue that her supposed difficulties (...) with atomism expose a deeper tension in her work between two fundamental metaphysical commitments each of which has substantial philosophical support: her monist theory of the material world (which maintains that there exists just one natural substance which is the single principal cause) and her occasional theory of causation (which requires multiple finite principal causes in nature -- causes that might be considered individual substances). Her monism undermines atomism while her theory of occasional cause seems to rest on a conception of nature that would be especially friendly to atomism. I argue further that we can solve this tension within a Cavendishian framework in such a way as to preserve her theory of causation and her monism, but that this solution depends upon our taking her monism in a particular (and weak) form. I finally note that we can best make sense of her unique and interesting form of monism by acknowledging her social-political motivations in addition to her motivations in natural philosophy. (shrink)

A comprehensive historical overview of formalist ideas in the philosophy of mathematics.

The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The (...) second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. -/- Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently. (shrink)

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

A long-standing theme in discussion of perception and thought has been that our primary cognitive contact with individual objects and events in the world derives from our perceptual contact with them. When I look at a duck in front of me, I am not merely presented with the fact that there is at least one duck in the area, rather I seem to be presented withthisthing (as one might put it from my perspective) in front of me, which looks to (...) me to be a duck. Furthermore, such a perception would seem to put me in a position not merely to make the existential judgment that there is some duck or other present, but rather to make a singular, demonstrative judgment, that that is a duck. My grounds for an existential judgment in this case derives from my apprehension of the demonstrative thought and not vice versa. (shrink)

I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not so novel as he (...) maintains. Finally, without attempting to give a full account of the notion of empirical mathematical evidence, I sketch a view showing how the use of calculation injects an empirical ingredient into proof. (shrink)

In this chapter, I examine similarities and divergences between Du Châtelet and Descartes on their endorsement of the use of hypotheses in science, using the work of Condillac to locate them in his scheme of systematizers. I conclude that, while Du Châtelet is still clearly a natural philosopher, as opposed to modern scientist, her conception of hypotheses is considerably more modern than is Descartes’, a difference that finds its roots in their divergence on the nature of first principles.

In this paper, I argue that Margaret Cavendish’s account of freedom, and the role of education in freedom, is better able to account for the specifics of women’s lives than are Thomas Hobbes’ accounts of these topics. The differences between the two is grounded in their differing conceptions of the metaphysics of human nature, though the full richness of Cavendish’s approach to women, their minds and their freedom can be appreciated only if we take account of her plays, accepting them (...) as philosophical texts alongside her more standard philosophical treatises. (shrink)

It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...) selecting beliefs. The significance of this question for assessing "intensional" results like Godel's Second Theorem, and their bearing on Hilbert's Program are discussed. (shrink)

It has often been noted that Margaret Cavendish discusses God in her writings on natural philosophy far more than one might think she ought to given her explicit claim that a study of God belongs to theology which is to be kept strictly separate from studies in natural philosophy. In this article, I examine one way in which God enters substantially into her natural philosophy, namely the role he plays in her particular version of teleology. I conclude that, while Cavendish (...) has some resources with which to partially alleviate this tension, she is nonetheless left with a significant difficulty. (shrink)

Whatever may be said about contemporary feminists’ evaluation of Descartes’ role in the history of feminism, Mary Astell herself believed that Descartes’ philosophy held tremendous promise for women. His urging all people to eschew the tyranny of custom and authority in order to uncover the knowledge that could be found in each one of our unsexed souls potentially offered women a great deal of intellectual and personal freedom and power. Certainly Astell often read Descartes in this way, and Astell herself (...) has been interpreted as a feminist – indeed, as the first English feminist. But a close look at Astell’s and Descartes’ theories of reason, and the role of authority in knowledge formation as well as in their philosophies of education, show that there are subtle yet crucial divergences in their thought – divergences which force us to temper our evaluation of Astell as a feminist. -/- My first task is to evaluate Astell’s views on custom and authority in knowledge formation and education by comparing her ideas with those of Descartes. While it is true that Astell seems to share Descartes’ wariness of custom and authority, a careful reading of her work shows that the wariness extends only as far as the tyranny of custom over individual intellectual development. It does not extend to a wariness about social and institutional customs and authority (including, perhaps most crucially, the institution of marriage as we see in her Reflection on Marriage). The reason for this is that Astell’s driving goal is to help women to come to know God’s plan for women – both in their roles as human and in their roles as women. According to Astell, while it is true that, as individuals, women must develop their rational capacities to the fullest in order to honor God and his plan for women as human, as members of social institutions, including the institution of marriage, women must subordinate themselves to men, including their husbands, in this case so as to honor God and his plan for women as women. Once we understand the theological underpinnings of her equivocal reaction to authority and custom, we can see that Astell may be considered a feminist in a very tempered way. -/- My second task is to use these initial conclusions to re-read her proposal for single-sexed education that we find in A Serious Proposal to the Ladies. It is true that Astell encourages women to join single-sexed educational institutions for the unique and empowering friendships that women can develop in such institutions. Still, my argument continues, the development of such friendships is not entirely an end in itself. Rather, Astell encourages women to develop such friendships such that they can re-enter the broader world armed with the tools that will help them endure burdensome features of the lives that await them in the world, including their lives as subordinated wives –burdens that Astell does not, in principle, challenge. (shrink)

As a practicing life scientist, Descartes must have a theory of what it means to be a living being. In this paper, I provide an account of what his theoretical conception of living bodies must be. I then show that this conception might well run afoul of his rejection of final causal explanations in natural philosophy. Nonetheless, I show how Descartes might have made use of such explanations as merely hypothetical, even though he explicitly blocks this move. I conclude by (...) suggesting that there is no reason for him to have blocked the use of hypothetical final causes in this way. (shrink)

Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have a (...) priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

The type of completeness Whitehead and Russell aimed for in their Principia Mathematica was what I call descriptive completeness. This is completeness with respect to the propositions that have been proved in traditional mathematics. The notion of completeness addressed by Gödel in his famous work of 1930 and 1931 was completeness with respect to the truths expressible in a given language. What are the relative significances of these different conceptions of completeness for traditional mathematics? What, if any, effects does incompleteness (...) of the type Gödel had in mind have for the evaluation of the Principia Mathematica project and other projects like it? (shrink)

This book defends antitheodicism, arguing that theodicies, seeking to excuse God for evil and suffering in the world, fail to ethically acknowledge the victims of suffering. The authors argue for this view using literary and philosophical resources, commencing with Immanuel Kant's 1791 "Theodicy Essay" and its reading of the Book of Job. Three important twentieth century antitheodicist positions are explored, including "Jewish" post-Holocaust ethical antitheodicism, Wittgensteinian antitheodicism exemplified by D.Z. Phillips and pragmatist antitheodicism defended by William James. The authors argue (...) that these approaches to evil and suffering are fundamentally Kantian. Literary works such as Franz Kafka'sThe Trial, Samuel Beckett's Waiting for Godot, and George Orwell's Nineteen Eighty-Four, are examined in order to crucially advance the philosophical case for antitheodicism. (shrink)

The theories of pre-existence and epigenesis are typically taken to be opposing theories of generation in the seventeenth and eighteenth centuries. One can be a pre-existence theorist only if one does not espouse epigenesis and vice versa. It has also been recognized, however, that the line between pre-existence and epigenesis in the nineteenth century, at least, is considerably less sharp and clear than it was in earlier centuries. The debate (1759-1777) between Albrecht von Haller and Caspar Friedrich Wolff on their (...) theories of generation is usually taken to be a debate between a pre-existence theorist and an epigeneticist, and the supposed fact that these two theories of generation are mutually exclusive explains (so the story goes) the divide between Haller and Wolff. However, it’s not clear that Haller endorsed an especially robust form of pre-existence, and nor is it clear that Wolff’s theory of generation -- once he considered it carefully -- is clear epigenetic. Rather, Haller’s theory of generation is marked by traces of epigensis and Wolff’s theory has elements of pre-existence theory. This is not to say that their theories of generation are basically the same, but the debate between the two ought not to be framed in terms of pre-existence versus epigenesis. Their points of difference must be explained in some other way. In this way, their controversy bears characteristics of similar disputes over generation in the nineteenth century more than it resembles those in the seventeenth century. In this paper, I argue that (a) Haller’s and Wolff’s theories both blend elements from pre-existence and epigenesis; (b) but there are still deeply-rooted differences between the two generation theories; (c) one source of these differences is that Haller and Wolff have divergent conceptions of what an adequate explanation is; (d) we can see that they have different conceptions of what constitutes an adequate explanation by paying heed to their evaluations of Descartes’ epistemology and methodology in his theory of generation (such as it is); and (e) this shows that one of the main differences between Haller’s generation theory and that of Wolff is the degree to which each thinks we need to (and indeed can) explain the nature of the causes -- both efficient and final -- at play during the formation of organic beings. (shrink)