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.

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)

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?

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)

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.

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

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)

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.

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)

This article describes a method of scoping for potential ethical contentions within a resource constrained research environment where actor participation and bottom–up analysis is precluded. Instead of reverting to a top–down analytical structure, a data-led process is devised. This imitates a bottom–up analytic structure in the absence of the direct participation of actors, culminating in the construction of a map of the ethical landscape; a high-resolution ethical matrix of coded interpretations of various actors’ ethical framings of the technology. Despite its (...) limitations, which are discussed, the map can subsequently support the identification of areas where ethical contentions may be raised. Here, the method is described with reference to the construction and analysis of a map of the ethical landscape of carbon capture and storage technology. Taken as a preliminary stage of a larger study, it can support the design and initiation of more sophisticated analyses which may integrate stronger bottom–up participation and facilitate a reflective, deliberative process amongst actors. (shrink)

This book presents an academic introduction, presentation and argument of stakeholder theory as as a model for strategic management of business firms and corporations and public organizations and institutions. The concept of stakeholder is generally used for the parties that affect or are affected by the activities of private or public organizations. Stakeholders are those interested parties who, other than shareholders, have a connection with the activities of a corporation, a firm or an organization. The reference to the stakeholders refers (...) to a conception of a business firm as founded on negotiated governance, where the maximization of the value for the shareholder is not the ultimate criterion. In this model the stakes and the interests that are not those of shareholders or investors, and which go beyond capital and include civil society, are important. This book presents this theory and makes it known as an ethical model for strategic management that is both concrete and useful for developing democracy in the firm, and making it possible to present elements of a social contract in the context of global capitalism. (shrink)

This chapter contains section titled: The Criteria of State Conduct Against Realism and Natural Law The Kantian Inheritance Rawls's Debt to Rousseau ‐ the Third Criterion Conclusion Notes.

In recent years there has been a noticeable yet largely unacknowledged ‘pragmatic turn’ in the scientific realism debate, inspired in part by van Fraassen’s work on ‘epistemic stances’. Features of this new approach include: an ascent to the meta-level (the focus is not so much on whether scientific realism is true, but on the prior questions of the nature of the positions in this debate, how to decide whether to be a scientific realist, etc.); a reinterpretation of scientific realism and (...) anti-realism as (or as closely associated with) stances or frameworks, rather than theories or beliefs; a move away from the previously dominant empirical-explanatory (i.e. quasi-scientific or naturalistic) conception of scientific realism, anti-realism, and their justification; and a stress on the pragmatic and values-based elements in the debate. The traditional scientific realism debate is concerned with determining which position is true, or most epistemically justified. The new approach by contrast is concerned with determining which position best serves certain values, e.g. is most useful, fruitful, or otherwise prudentially preferable. In this paper we try to bring together the various strands in this new orientation, summarise its key features, contrast it with superficially similar but opposing views, and explore the similarities and differences among some of its adherents. Given we are advocates of the turn, we also offer a defence of the value and fruitfulness of this reconceptualization of the debate. (shrink)

This paper examines the idea of objective phenomenology, or a way of understanding the phenomenal character of conscious experiences that doesn’t require one to have had the kinds of experiences under consideration. My central thesis is that structural facts about experience—facts that characterize purely how conscious experiences are structured—are objective phenomenal facts. I begin by precisifying the idea of objective phenomenology and diagnosing what makes any given phenomenal fact subjective. Then I defend the view that structural facts about experience are (...) objective. I also argue that structural facts about experience, despite being objective, nevertheless still give rise to an explanatory gap. (shrink)

The Routledge Handbook of Collective Intelligence for Democracy and Governance explores the concepts, methodologies, and implications of collective intelligence for democratic governance, in the first comprehensive survey of this field. Illustrated by a collection of inspiring case studies and edited by three pioneers in collective intelligence, this handbook serves as a unique primer on the science of collective intelligence applied to public challenges and will inspire public actors, academics, students, and activists across the world to apply collective intelligence in policymaking (...) and administration to explore its potential, both to foster policy innovations and reinvent democracy. The Routledge Handbook of Collective Intelligence for Democracy and Governance is essential reading and an authoritative reference for scholars, students, researchers and practitioners of public policy, public administration, governance, public management, information technology and systems, innovation and democracy as well as more broadly for political science, psychology, management studies, public organizations and individual policy practitioners, public authorities, civil society activists and service providers. (shrink)

Dismissed as a miserable elitist who condemned popular culture in the name of 'high art', Theodor W. Adorno (1903–1969) is one of the most provocative and important yet least understood of contemporary thinkers. This book challenges this popular image and re-examines Adorno as a utopian philosopher who believed authentic art could save the world. Adorno Reframed is not only a comprehensive introduction to the reader coming to Adorno for the first time, but also an important re-evaluation of this founder of (...) the Frankfurt School. Using a wealth of concrete illustrations from popular culture, Geoffrey Boucher recasts Adorno as a revolutionary whose subversive irony and profoundly historical aesthetics defended the integrity of the individual against social totality. (shrink)

Some accounts of moral responsibility hold that an agent's responsibility is completely determined by some aspect of the agent's mental life at the time of action. For example, some hold that an agent is responsible if and only if there is an appropriate mesh among the agent's particular psychological elements. It is often objected that the particular features of the agent's mental life to which these theorists appeal (such as a particular structure or mesh) are not necessary for responsibility. This (...) is because there appear to be cases in which an agent acts at an earlier time which causes her to lack the appropriate psychological features at some later time and yet, intuitively, she is responsible at that later .. (shrink)

Perception is riddled with cases of reliable misperception. These are cases in which a perceptual state is tokened inaccurately any time it is tokened under normal conditions. On the face of it, this fact causes trouble for theories that provide an analysis of perceptual content in non-semantic, non-intentional, and non-phenomenal terms, such as those found in Millikan (1984), Fodor (1990), Neander (2017), and Schellenberg (2018). I show how such theories can be extended so that they cover such cases without giving (...) up their core commitments. (shrink)

We are all familiar with the phenomenon of a proposition seeming true. Many think that these seeming states can yield justified beliefs. Very few have seriously explored what these seeming states are. I argue that seeming states are not plausibly analyzed in terms of beliefs, partial beliefs, attractions to believe, or inclinations to believe. Given that the main candidates for analyzing seeming states are unsatisfactory, I argue for a brute view of seemings that treats seeming states as irreducible propositional attitudes.

This is a discussion of the state of discussion on pragmatic reasons for belief.