These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.

For those who have understood the solution to the Liarʼs Paradox and the Paradoxes of Predication, presented in A Comprehensive Solution to the Paradoxes and The Solution to the Liarʼs Paradox1, it will come as no surprise how the Berry Paradox should be solved. Nonetheless, the solution will be presented here in a short note, for completenessʼ sake.

Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.

I recently had the occasion to reread Naming and Necessity by Saul Kripke. NaN struck me this time, as it always has, as breathtakingly clear and lucid. It also struck me this time, as it always has, as wrong-headed in several major ways, both in its methodology and its content. Herein is a brief explanation why.

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...) Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

Because it is time-dependent, parallel computation is fundamentally different from sequential computation. Parallel programs are non-deterministic and are not effective procedures. Given the brain operates in parallel, this casts doubt on AI's attempt to make sequential computers intelligent.

A solution to the Liar must do two things. First, it should say exactly which step in the Liar reasoning - the reasoning which leads to a contradiction - is invalid. Secondly, it should explains why this step is invalid.

I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...) there to hear it, then the answer to the question is evidently "no." But if noise means that, if there were (counterfactually) someone there, then he would hear it, then just as obviously, the answer becomes "yes.". (shrink)

It might seem that three of Godel’s results - the Completeness and the First and Second Incompleteness Theorems - assume so little that they are reasonably indisputable. A version of the Completeness Theorem, for instance, can be proven in RCA0, which is the weakest system studied extensively in Simpson’s encyclopaedic Subsystems of Second Order Arithmetic. And it often seems that the minimum requirements for a system just to express the Incompleteness Theorems are sufficient to prove them. However, it will be (...) shown that a particular sub-system of Peano Arithmetic is powerful enough to express assertions about syntax, provability, consistency, and models, while being too weak to allow the standard proofs of the theorems to go through. An alternative proof is available for the First Incompleteness Theorem, but is of such a different nature that the import of the theorem changes. And there are no alternative proofs for (certainly) the Completeness and (apparently) the Second Incompleteness Theorems. It is therefore perfectly rational for someone to be skeptical about Godel’s results. (shrink)

The system called F is essentially a sub-theory of Frege Arithmetic without the ad infinitum assumption that there is always a next number. In a series of papers (Systems for a Foundation of Arithmetic, True” Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity) it was shown that F proves a large number of basic arithmetic truths, such as the Euclidean Algorithm, Unique Prime Factorization (i.e. the Fundamental Law of Arithmetic), and Quadratic Reciprocity, indeed a sizable amount of arithmetic. (...) In particular, F proves some (but not all) of the Peano Axioms; that is, F proves the axioms of a sub-theory - call it FPA - of second-order Peano-Arithmetic. This short technical note will demonstrate that the converse also holds, in the following sense. F has the same language as second-order Peano Arithmetic except that, in addition, it has a two-place predicate symbol “Μ”. Then it is possible to provide a definition, indeed a reasonable definition, for “Μ” such that FPA proves all the axioms of F. So F and FPA effectively have the same proof-theoretic strength. In particular FPA, which lacks the Successor Axiom stating that every natural number has a successor, is able to prove the Euclidean Algorithm, Unique Prime Factorization, and Quadratic Reciprocity, indeed (again) a sizable amount of arithmetic. (shrink)

In "The Nature and Meaning of Numbers," Dedekind produces an original, quite remarkable proof for the holy grail in the foundations of elementary arithmetic, that there are an infinite number of things. It goes like this. [p, 64 in the Dover edition.] Consider the set S of things which can be objects of my thought. Define the function phi(s), which maps an element s of S to the thought that s can be an object of my thought. Then phi is (...) evidently one-to-one, and the image of phi is contained in S. Indeed, it is properly contained in S, because I myself can be an object of my thoughts and so belong to S, but I myself am not a mere thought. Thus S is infinite. (shrink)

How-many numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed how-many number in terms of one-to-one correspondence between two sets. That is to say, one arrived at numbers by either abstracting from the concept of correspondence, in the case of Cantor, or by using it to provide an out-and-out definition, in the case of Frege.

A solution to the paradoxes has two sides: the philosophical and the technical. The paradoxes are, first and foremost, a philosophical problem. A philosophical solution must pinpoint the exact step where the reasoning that leads to contradiction is fallacious, and then explain why it is so.

As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject matter and its study (...) is a serious one, because in the end, one can lose sight of what one should be doing in the first place. Perhaps it is taking matters too literally, but it seems that there is something to be said for taking the term to represent X. Doing so and accepting the term to have some kind of significance, it is then natural to focus on the question of what a foundation of arithmetic should be; and, if one exists, what one is. Whatever the case, that is what shall be done in this paper. (shrink)

On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?

Andrew Marvell’s “To His Coy Mistress” is one of the best known and most commented on poems in the English language. According to the critical consensus, the poem is a seduction gambit in the “Carpe Diem” tradition. Interpretive debate therefore revolves around the significance of the allusions and imagery of the poem, rather than its central meaning. Moving against the current, this article challenges the critical consensus that Marvell’s “To His Coy Mistress” is a poem that has seduction as (...) its main significance or implied intention. Reading the poem with attention to its ironic moments and theological references reveals that its allusions and imagery are systematically ambivalent. In the context of Marvell’s other poetry, especially “Dialogue of the Soul and the Body,” it becomes possible to show that “Coy Mistress” shares many features with these metaphysical meditations on mortality and spirituality. By making reference to psychoanalytic theory, the article then demonstrates the plausibility of a reading in which the poem aims to avoid, rather than engorge, sexual desire. The poem is a monument to repression, and a reminder of mortality, not a love lyric. (shrink)

The extent to which British Idealism was heavily influenced by Scots has been little noticed, yet not only were they at the forefront of introducing Hegel into Britain in the work of Ferrier, Carlyle, Hutcheson, Stirling and Edward Caird, but they were also distinctive in locating themselves in relation to the Scottish philosophical tradition they sought to extend. The Scottish Idealists, among them Edward Caird, David George Ritchie, Andrew Seth Pringle Pattison, William Mitchell, John Watson, and the Welshman Henry (...) Jones who found his spiritual home in Glasgow, comprised a formidable force and dominated the philosophical professoriate in Britain, Australia and Canada from the late nineteenth century to the years leading up to the First World War. Its main centres were St. Andrews, Glasgow and Edinburgh in Scotland, Cardiff in Wales, and Oxford in England. This collection of readings, the first of its kind, has been chosen with a view to displaying the variety, richness and strength of the Scottish Idealist tradition, beginning with an essay from the famous Essays in Philosophical Criticism, a book that set-out the future direction of enquiry for this group of thinkers who shared a 'common purpose or tendency'. Scottish Idealism was immensely spiritual in character and recognized no hard and fast distinctions between philosophy, religion, poetry and science. It was a formidable force in social and educational reform.. (shrink)

This paper explores the rhetoric of obstetric ultrasound technology as it relates to the abortion debate, specifically the interpretation given to ultrasound images by opponents of abortion. The tenor of the anti-abortion approach is precisely captured in the videotape, Ultrasound:A Window to the Womb. Aspects of this videotape are analyzed in order to tease out the assumptions about the (female) body and about the access to truth yielded by scientific technology (ultrasound) held by militant opponents of abortion. It is argued (...) that the ultrasound images do not offer transparent confirmation of the ontological status of the embryo and fetus. Rather, the window of ultrasound is constructed through a complex combination of visual and verbal devices: ultrasound images, photographic images, verbal argument, and emotional appeal. (shrink)

The first edition was awarded the '2005 Textbook Award of the Association of University Professors of Management (Verband der Hochschullehrer fur ...

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...) to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks. (shrink)

We all face death, but how many of us are actually ready for it? Whether our own death or that of a loved one comes first, how prepared are we, spiritually or practically? In Preparing to Die, Andrew Holecek presents a wide array of resources to help the reader address this unfinished business. Part One shows how to prepare one's mind and how to help others, before, during, and after death. The author explains how spiritual preparation for death can (...) completely transform our relationship to the end of life, dissolving our fear and helping us to feel open and receptive to letting go in the dying process. Daily meditation practices, the stages of dying and how to work with them, and after-death experiences are all detailed in ways that will be particularly helpful for those with an interest in Tibetan Buddhism and in Tibetan approaches to conscious dying. Part Two addresses the practical issues that surround death. Experts in grief, hospice, the funeral business, and the medical and legal issues of death contribute chapters to prepare the reader for every practical concern, including advance directives, green funerals, the signs of death, warnings about the funeral industry, the stages of grief, and practical care for the dying. Part Three contains heart-advice from twenty of the best-known Tibetan Buddhist masters now teaching in the West. These brief interviews provide words of solace and wisdom to guide the dying and their caregivers during this challenging time. Preparing to Die is for anyone interested in learning how to prepare for death from a Buddhist perspective, both spiritually and practically. It is also for those who want to learn how to help someone else who is dying, both during the time of illness and death as well as after death. (shrink)

How should politicians act? When should they try to lead public opinion and when should they follow it? Should politicians see themselves as experts, whose opinions have greater authority than other people's, or as participants in a common dialogue with ordinary citizens? When do virtues like toleration and willingness to compromise deteriorate into moral weakness? In this innovative work, Andrew Sabl answers these questions by exploring what a democratic polity needs from its leaders. He concludes that there are systematic, (...) principled reasons for the holders of divergent political offices or roles to act differently. Sabl argues that the morally committed civil rights activist, the elected representative pursuing legislative results, and the grassroots organizer determined to empower ordinary citizens all have crucial democratic functions. But they are different functions, calling for different practices and different qualities of political character. To make this case, he draws on political theory, moral philosophy, leadership studies, and biographical examples ranging from Everett Dirksen to Ella Baker, Frances Willard to Stokely Carmichael, Martin Luther King Jr. to Joe McCarthy. Ruling Passions asks democratic theorists to pay more attention to the "governing pluralism" that characterizes a diverse, complex democracy. It challenges moral philosophy to adapt its prescriptions to the real requirements of democratic life, to pay more attention to the virtues of political compromise and the varieties of human character. And it calls on all democratic citizens to appreciate "democratic constancy": the limited yet serious standard of ethical character to which imperfect democratic citizens may rightly hold their leaders--and themselves. (shrink)

Première introduction au séminaire de 1978-1979 de Michel Foucault au Collège de France. Depuis la mort de Michel Foucault en 1984, de nombreuses exégèses ou critiques de sa pensée ont été écrites. La publication en 1999 de Dits et Ecrits a suscité de nouvelles interprétations des textes de Foucault mettant son travail en perspective. Ainsi de l'éthique et de la biopolitique. Cependant à isoler éthique et biopolitique, on risquerait de ne pas prendre la mesure de l'apport de Foucault. Peut-on en (...) effet rendre autonomes l'éthique et la question de la subjectivation d'une philosophie politique? Peut-on comprendre le concept de biopolitique sans le relier à sa réflexion sur le libéralisme? La biopolitique, selon Foucault, est la technologie politique du libéralisme naissant. Elle vise à améliorer ses performances jusqu'au déclin de sa raison d'être. Un libéralisme sans liberté interroge l'obstination avec laquelle Foucault tente de proposer de nouvelles catégories de philosophie politique qui ne soient pas l'exclusive d'une philosophie du pouvoir. En effet, l'introduction du terme de libéralisme est concomitante de la création de concepts comme ceux de biopolitique, de gouvernementalité. L'instauration de ces notions est complexe car elle renvoie à une lecture inédite du libéralisme qui n'est pas considérée comme une doctrine, ni comme une construction d'un système dont la liberté serait le centre, non plus qu'un idéal politique, voire une idéologie dominante. Pour Foucault, il s'agit d'une effectuation d'un type de rationalité qui met en œuvre une technologie politique spécifique. (shrink)

This paper argues that the dominance of linear models has led many sociologists to construe the social world in terms of a "general linear reality." This reality assumes (1) that the social world consists of fixed entities with variable attributes, (2) that cause cannot flow from "small" to "large" attributes/events, (3) that causal attributes have only one causal pattern at once, (4) that the sequence of events does not influence their outcome, (5) that the "careers" of entities are largely independent, (...) and (6) that causal attributes are generally independent of each other. The paper discusses examples of these assumptions in empirical work, consider standard and new methods addressing them, and briefly explores alternative models for reality that employ demographic, sequential, and network perspectives. (shrink)

A Physicalist Manifesto is a full treatment of the comprehensive physicalist view that, in some important sense, everything is physical. Andrew Melnyk argues that the view is best formulated by appeal to a carefully worked-out notion of realization, rather than supervenience; that, so formulated, physicalism must be importantly reductionist; that it need not repudiate causal and explanatory claims framed in non-physical language; and that it has the a posteriori epistemic status of a broad-scope scientific hypothesis. Two concluding chapters argue (...) in detail that contemporary science provides no significant empirical evidence against physicalism and some considerable evidence for it. Written in a brisk, candid and exceptionally clear style, this 2003 book should appeal to professionals and students in philosophy of mind, metaphysics and philosophy of science. (shrink)

Empirical work has lately confirmed what many philosophers have taken to be true: people are ‘biased toward the future’. All else being equal, we usually prefer to have positive experiences in the future, and negative experiences in the past. According to one hypothesis, the temporal metaphysics hypothesis, future-bias is explained either by our beliefs about temporal metaphysics—the temporal belief hypothesis—or alternatively by our temporal phenomenology—the temporal phenomenology hypothesis. We empirically investigate a particular version of the temporal belief hypothesis according to (...) which future-bias is explained by the belief that time robustly passes. Our results do not match the apparent predictions of this hypothesis, and so provide evidence against it. But we also find that people give more future-biased responses when asked to simulate a belief in robust passage. We take this to suggest that the phenomenology that attends simulation of that belief may be partially responsible for future-bias, and we examine the implications of these results for debates about the rationality of future-bias. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs (...) of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

Empirical evidence suggests that a majority of people believe that time robustly passes, and that many also report that it seems to them, in experience, as though time robustly passes. Non-dynamists deny that time robustly passes, and many contemporary non-dynamists—deflationists—even deny that it seems to us as though time robustly passes. Non-dynamists, then, face the dual challenge of explaining why people have such beliefs and make such reports about their experiences. Several philosophers have suggested the stable-self explanation, according to which (...) what partly explains one or both of the explananda is that people believe themselves to have a stable persisting self which is imagined either as moving through time, or as being stationary, with time moving around it. This paper empirically investigates this explanation. We found, however, no evidential support for the stable-self explanation, leaving the non-dynamist but also, we will argue, the dynamist, with an extant explanatory challenge. (shrink)