In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

We examine the notions of negative, infinite and hotter than infinite temperatures and show how these unusual concepts gain legitimacy in quantum statistical mechanics. We ask if the existence of an infinite temperature implies the existence of an actual infinity and argue that it does not. Since one can sensibly talk about hotter than infinite temperatures, we ask if one could legitimately speak of other physical quantities, such as length and duration, in analogous terms. That is, could there be longer (...) than infinite lengths or temporal durations? We argue that the answer is surprisingly yes, and we outline the properties of a number system that could be employed to characterize such magnitudes. (shrink)

Since Euclid defined a point as “that which has no part” it has been widely assumed that points are necessarily unextended. It has also been assumed that this is equivalent to saying that points or, more properly speaking, degenerate segments, have length zero. We challenge these assumptions by providing models of Euclidean geometry where the points are extended despite the fact that the degenerate segments have null lengths, and observe that whereas the extended natures of the points are not recognizable (...) in the given models, they can be recognized and characterized by structures that are suitable expansions of the models. (shrink)

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\mathbf {No}}$, i.e. a subfield of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$. In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of (...) Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $$. These include all models of $T$, where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$, which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R})^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $. (shrink)

We suggest that, far from establishing an inconsistency in the standard theory of the geometrical linear continuum, Zeno’s Paradox of Extension merely establishes an inconsistency between the standard theory of geometrical magnitude and a misguided system of length measurement. We further suggest that our resolution of Zeno’s paradox is superior to Adolf Grünbaum’s now standard resolution based on Lebesgue measure theory.

Novel (categorical) axiomatizations of the classical arithmetic and geometric continua are provided and it is noted that by simply deleting the Archimedean condition one obtains (categorical) axiomatizations of J.H. Conway's ordered field No and its elementary n-dimensional metric Euclidean, hyperbolic and elliptic geometric counterparts. On the basis of this and related considerations it is suggested that whereas the classical arithmetic and geometric continua should merely be regarded as arithmetic and geometric continua modulo the Archimedean condition, No and its geometric counterparts (...) may be regarded as absolute arithmetic and geometric continua modulo von Neumann-Bernays-Godel set theory. (shrink)

The official model of explanation proposed by the logical empiricists, the covering law model, is subject to familiar objections. The goal of the present paper is to explore an unofficial view of explanation which logical empiricists have sometimes suggested, the view of explanation as unification. I try to show that this view can be developed so as to provide insight into major episodes in the history of science, and that it can overcome some of the most serious difficulties besetting the (...) covering law model. (shrink)

In this work, the prominent political philosopher Philip Pettit embarks on a massive undertaking to offers major new accounts of the foundations of the state and the nature of justice. In doing so Pettit builds a new theory of what the state is and what it ought to be, addresses the normative question of how justice serves as a measure of the success of a state, and the way it should operate in relation to its citizens and other people.

Die Entwicklung eines philosophischen Systems ist für den Autor Folge der persönlichen Auseinandersetzung mit der Frage nach dem Lebenssinn und der Stellung des Menschen in der Welt. Die Frage nach dem Sinn des menschlichen Lebens ist die zentrale Frage des Menschen. Die zentrale Stellung bekommt diese Frage dadurch, dass der Mensch selbst - mehr unbewusst als bewusst - sie in seinem Verhalten, seinen Absichten und in seiner Lebensweise, in dem, was er vom Leben im Allgemeinen und von seinem persönlichen Leben (...) im Besonderen fordert und in ihm erreichen will, beantwortet. Diese Frage wird durch die Möglichkeit eines konkret bestimmten individuellen Lebensvollzugs gestellt und durch den Vollzug dieses individuellen, persönlichen Lebens beantwortet. Das eigentliche Problem eines jeden einzelnen Menschen ist seine Orientierung in der Welt. Den «Kompass» für die Orientierung in der Welt kann er nur in der Erkenntnis der Wirklichkeit finden. (shrink)

A person presented with adequate but not conclusive evidence for a proposition is in a position voluntarily to acquire a belief in that proposition, or to suspend judgment about it. The availability of doxastic options in such cases grounds a moderate form of doxastic voluntarism not based on practical motives, and therefore distinct from pragmatism. In such cases, belief-acquisition or suspension of judgment meets standard conditions on willing: it can express stable character traits of the agent, it can be responsive (...) to reasons, and it is compatible with a subjective awareness of the available options. (shrink)

How Galileo created the problem of consciousness -- Is there a ghost in the machine? -- Can physical science explain consciousness? -- How to solve the problem of consciousness -- Consciousness and the meaning of life.

What's the Use of Philosophy? aims to answer the question posed in its title, whether the questioner intends to dismiss philosophy, or seeks a positive answer. The first three chapters explore the grounds for dismissal. Chapter 1 expresses skepticism about the value of much professional Anglophone philosophy, while recognizing virtues in work often viewed as peripheral. Chapter 2 studies a philosophical subfield, the philosophy of science, arguing that, while its condition may be better than the norm, it is far from (...) perfect. In Chapter 3, a more systematic critique is offered, in the form of a pathology report. Its diagnosis identifies six justifiable philosophical impulses that, when pursued obsessively, generate distortions and diseases: clarity becomes a fetish, formalism becomes obfuscation, "intuition" is given burdens it cannot bear, appeals to the a priori sprinkle fairy dust on controversial claims, ideas grabbed from other fields are used in cartoon forms, and questions no longer relevant are treated as if they were "timeless." Chapter 4 attempts to identify the legitimate goals of philosophy. It sees value in attempts to respond to questions arising from the difficulties of the age, in attempts to reform methods of inquiry, and, above all, in the construction of syntheses capable of helping people as they struggle to make sense of their lives. Philosophy is renewed when these traditional goals once again become central to its discussions. The final chapter responds to a frequently voiced worry in a letter to young philosophers who wonder how they can align their aspirations with the hyper-professionalism expected of them. (shrink)

Placebo cures. Global warming. Extraterrestrial life. Psychokinesis. In a time when scientific claims can sound as strange as science fiction--and can have a profound effect on individual life or public policy--assessing the merits of a far-out, supposedly scientific idea can be as difficult as it is urgent. Into the breach between helpless gullibility and unyielding skepticism steps physicist Robert Ehrlich, with an indispensable guide to making sense of "scientific" claims. A series of case studies of some of the most (...) controversial (and for the judging public, deeply vexing) topics in the natural and social sciences, Ehrlich's book serves as a primer for evaluating the evidence for the sort of strange-sounding ideas that can shape our lives. A much-anticipated follow-up to his popular Nine Crazy Ideas in Science, this book takes up issues close to readers' everyday reality--issues such as global warming, the dangers of cholesterol, and the effectiveness of placebos--as well as questions that resonate through (and beyond) civic life: Is intelligent design a scientific alternative to evolution? Is homosexuality primarily innate? Are people getting smarter or dumber? In each case, Ehrlich shows readers how to use the tools of science to judge the accuracy of strange ideas and the trustworthiness of ubiquitous "experts." As entertaining as it is instructive, his book will make the work of living wisely a bit easier and more reliable for scientists and nonscientists alike. (shrink)

Engineers are traditionally regarded as trustworthy professionals who meet exacting standards. In this chapter I begin by explicating our trust relationship towards engineers, arguing that it is a linear but indirect relationship in which engineers “stand behind” the artifacts and technological systems that we rely on directly. The chapter goes on to explain how this relationship has become more complex as engineers have taken on two additional aims: the aim of social engineering to create and steer trust between people, and (...) the aim of creating automated systems that take over human tasks and are meant to invite the trust of those who rely on and interact with them. (shrink)

An esteemed philosopher discusses his theory of universal freedom, describing how even those who are members of free societies may find their liberties curtailed and includes tests of freedom including the eyeball test and the tough-luck test.

This paper argues for a Husserlian account of phenomenal intentionality. Experience is intentional insofar as it presents a mind-independent, objective world. Its doing so is a matter of the way it hangs together, its having a certain structure. But in order for the intentionality in question to be properly understood as phenomenal intentionality, this structure must inhere in experience as a phenomenal feature. Husserl’s concept of horizon designates this intentionality-bestowing experiential structure, while his concept of motivation designates the unique phenomenal (...) character of this structure as it is experientially lived through. The way experience hangs together is itself a phenomenal feature of experience. (shrink)

Good reasoning can lead to success; bad reasoning can lead to catastrophe. Yet, it's not obvious how we reason, and why we make mistakes. This new book by one of the pioneers of the field, Philip Johnson-Laird, looks at the mental processes that underlie our reasoning. It provides the most accessible account yet of the science of reasoning.

This paper has two goals: (1) to defend a particular response to the problem of resultant moral luck and (2) to defend the claim that we are only responsible for what we could have avoided. Cases of overdetermination threaten to undermine the claim that we are only responsible for what we could have avoided. To deal with this issue, I will motivate a particular way of responding to the problem of resultant moral luck. I defend the view that one's degree (...) of responsibility is immune to moral luck but the scope of events for which one is responsible is subject to moral luck. I will then argue that this view allows us to explain away certain intuitions about responsibility and overdetermination (as well as similar intuitions regarding Frankfurt Cases). As a result, the claim that we are only responsible for what we could have avoided becomes significantly more plausible. (shrink)

Cognitive neuropsychiatry (CN) is the explanation of psychiatric disorder by the methods of cognitive neuropsychology. Within CN there are, broadly speaking, two approaches to delusion. The first uses a one-stage model, in which delusions are explained as rationalizations of anomalous experiences via reasoning strategies that are not, in themselves, abnormal. Two-stage models invoke additional hypotheses about abnormalities of reasoning. In this paper, I examine what appears to be a very strong argument, developed within CN, in favor of a twostage explanation (...) of the difference in content between the Capgras and Cotard delusions. That explanation treats them as alternative rationalizations of essentially the same phenomenology. I show, however, that once we distinguish the phenomenology (and the neuroetiology), a one-stage model is adequate. In the final section I make some more general remarks on the oneand two-stage models. (shrink)