Despite the widespread influence of Nietzsche’s iconoclastic works on twentieth-century intellectual and literary culture, there has been a surprising lack of academic focus on the parallels between his philosophical trajectory and the anthropological literature on the mysterious god Dionysus. The striking correlations between ancient Dionysian worship with Nietzsche’s oeuvre revalorise the celebration of ecstasy. In Twilight of the Idols, he states that the Dionysian is “beyond pity and terror, to realize in oneself the eternal joy of becoming.” By elucidating some (...) of the mysteries surrounding this enigmatic god, this article aims to re-establish the central importance of Dionysus in Nietzsche’s work, and to suggest that if we are to understand Nietzsche at all, we must first undertake an anthropological reading of Dionysus. (shrink)

Presents a plethora of approaches to developing human potential in areas not conventionally addressed. Organized in two parts, this international collection of essays provides viable educational alternatives to those currently holding sway in an era of high-stakes accountability.

Enth.: Dugoald Stewart's account of Adam Smith / ed. by I.S. Ross.

Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical knowledge. A third is to serve mathematics, (...) by lending insight into the various fields and suggesting fruitful techniques of research. A fourth purpose of a foundation is to provide an arena for exploring relations and interactions between mathematical fields. While set theory does better with regard to some of these and worse with regard to others, it has become the de facto arena for deciding questions of existence, something one might expect of a foundation. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...) rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) slogan that mathematics is the science of structure? (shrink)

According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea (...) of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account. (shrink)

The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)

The articles in this volume represent a part of the philosophical literature on higher-order logic and the Skolem paradox. They ask the question what is second-order logic? and examine various interpretations of the Lowenheim-Skolem theorem.

At the dawn of Europe, an extraordinary Greek philosopher-engineer--biologist named Aristotle strove to develop a code that the good person could live by. Aristotle's concepts have much resonance with us today. Although there are differences in the way in which we perceive things, basically we all share the same aspirations and needs.This new edition will be of value to the globally-focused engineer and scientist; it includes case studies from both developed and developing nations to help students and practitioners unravel the (...) complexities of ethical practice in the 21st Century. It is testament to the global and indefatigable nature of our scientific community that 2,300 years after Aristotle, we might derive the same conclusions as he did. (shrink)

Managing Complexity in Healthcare introduces the ComEntEth (Complex Entropic Ethical) model as an integrated bio-medical and philosophical approach to understanding how people get things done in healthcare. Drawing on the complexity sciences, studies of entropy in living organisms, and the ethics of Emmanuel Levinas, healthcare is theorised as energetic relational exchanges between people as entropic and ethical entities that unfold around a central attractor: Reduction in elevated entropy or suffering in patients. Living entities are engaged in a continuous struggle against (...) the tendency to produce entropy. From the cellular to the collective of human endeavours, the tendency of complex systems is to disorder and decay. Yet in the micro-activity of healthcare enterprise, people resist this tendency by expending energy to create order and sustain life. Making sense of how this miraculous work is made possible is the foundation of this book. Through practical examples - from analysis of practitioner burnout, rural and remote healthcare, the functioning of emergency departments, to government, social and institutional responses to the COVID-19 pandemic -this new integral philosophy provides practitioners, managers, policy designers, and scholars an effective way to understand the dynamics of daily processes and practices that link the micro of everyday interactions with the macro-trends of healthcare. (shrink)

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...) when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’. (shrink)

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

The harm principle, understood as the normative requirement that conduct should be criminalized only if it is harmful, has difficulty in dealing with those core cases of criminal wrongdoing that can occur without causing any direct harm. Advocates of the harm principle typically find it implausible to hold that these core cases should not be crimes and so usually seek out some indirect harm that can justify criminalizing the seemingly harmless conduct. But this strategy justifies criminalization of a wide range (...) of conduct on the basis of the fear, worry, and anxiety it generates among those who are not the direct victims of the conduct, and thereby undermines the limiting role of the harm principle by permitting the very move it was meant to prevent: the criminalization of harmless conduct on the ground of others’ feelings about it. The best way to avoid this dilemma is to recognize that people have rights, operating independently of the harm principle, to be treated in certain ways just because they are persons. The existence of such rights provides a ground for both criminalizing conduct and limiting the scope of criminalization because these rights point both to conduct that people must be permitted to engage in and conduct that might well be criminalized. A complete account of criminal law will therefore require the harm principle to work together with an independent account of rights. (shrink)

A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...) is invoked along the way, suitably modified for the present anti-foundationalist setting. (shrink)

The purpose of this paper is to present a thought experiment and argument that spells trouble for “radical” deflationism concerning meaning and truth such as that advocated by the staunch nominalist Hartry Field. The thought experiment does not sit well with any view that limits a truth predicate to sentences understood by a given speaker or to sentences in (or translatable into) a given language, unless that language is universal. The scenario in question concerns sentences that are not understood but (...) are known to be logical consequences of known and understood sentences. Ultimately, the issue turns on the notion of logical consequence that is available to various versions of deflationism. (shrink)

In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One main theme of my (...) ante rem structuralism is that any coherent axiomatization describes a structure, or a class of structures. If one weakens the logic, then more axiomatizations become coherent. (shrink)

This chapter provides an overview of second-order logic and higher-order logic generally. It provides the basic formal languages, deductive systems, and model-theoretic semantics, including a brief account of George Boolos’s interpretation of second-order languages in terms of the plural construction. It then goes into some of the arguments in favor of second-order logic.

Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then (...) the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist. (shrink)

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. (...) We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic, in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally. (shrink)

The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.