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.

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.

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.

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)

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.

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)

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)

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)

One of the key achievements of critical realism has been to expose the modernist myth of universal reason, which holds that authentic knowledge claims must be objectively ‘pure’, uncontaminated by the subjectivity of local place, specific time and particular culture. Wright aims to address the lack of any substantial and sustained engagement between critical realism and theological critical realism with particular regard to: (a) the distinctive ontological claims of Christianity; (b) their epistemic warrant and intellectual legitimacy; and (c) scrutiny of (...) the primary source of the ontological claims of Christianity, namely the historical figure of Jesus of Nazareth. As such, it functions as a prolegomena to a much needed wider debate, guided by the under-labouring services of critical realism, between Christianity and various other religious and secular worldviews. This important new text will help stimulate a debate that has yet to get out of first gear. This book will appeal to academics, graduate and post-graduate students especially, but also Christian clergy, ministers and informed laity, and members of the general public concerned with the nature of religion and its place in contemporary society. (shrink)

Field theories have been central to physics over the last 150 years, and there are several theories in contemporary physics in which physical fields play key causal and explanatory roles. This paper proposes a novel field trope-bundle (FTB) ontology on which fields are composed of bundles of particularized property instances, called tropes and goes on to describe some virtues of this ontology. It begins with a critical examination of the dominant view about the ontology of fields, that fields are properties (...) of a substantial substratum. (shrink)

This book offers an exciting re-interpretation of Auguste Comte, the founder of French sociology. Following the development of his philosophy of positivism, Comte later focused on the importance of the emotions in his philosophy resulting in the creation of a new religious system, the Religion of Humanity. Andrew Wernick provides the first in-depth critique of Comte's concept of religion and its place in his thinking on politics, sociology and philosophy of science. He places Comte's ideas in the context of (...) post-1789 French political and intellectual history, and of modern philosophy, especially postmodernism. Wernick relates Comte to Marx and Nietzsche as seminal figures of modernity and examines key features of modern and postmodern French social theory, tracing the inherent flaws and disintegration of Comte's system. Wernick offers original and fascinating insights in this rich study which will attract a wide audience from sociologists and philosophers to cultural theorists and historians. (shrink)

The purpose of this article is to consider Collingwood’s Idea of History in the wider context of his thoughts on historical knowledge, and in the light of criticisms which have often been less than generous in giving a certain latitude to what he meant to convey. The article shows how the main doctrines, that are often taken in isolation and forensically analysed and criticized, may be defended and made more intelligible when considered as an integrated whole. Such an idea as (...) re-enactment, for example, need not be considered to lend itself to intuitionism in philosophy, nor methodological individualism in interpretation. It is possible to understand Collingwood presenting the theory of re-enactment as at once a method for the attainment of historical knowledge, which doesn’t simply come about fortuitously, while also being an ontological condition of understanding, that is, when historical knowledge is attained, something will have happened to us, that is, we will have thought exactly the same thoughts, broadly speaking, as the historical subjects themselves. (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)

In this paper I explore the pedagogical and political shift marked by the meaning and practice of diversity offered through New Labour education policy texts, specifically, the policy and practice of personalized learning (or personalization). The aim of this paper is to map the ways in which diversity relays and mobilizes a set of neoliberal positions and relationships in the field of education and seeks to govern education institutions and education users through politically circulating norms and values. These norms and (...) values, I want to argue, echo and redeem the kinds of frameworks, applications and rationalities typically aligned with modes of neoliberal or advanced liberal governance, e.g. marketization, monetarization, atomization and deregulation. I conclude the paper by considering how diversity in education renders problematic conventional antinomies of the citizen and consumer, public and private, state and civil society, etc., and forces us to confront the rhizomatic character of contemporary governance and education in the era of neoliberalism. (shrink)

Baghdad, California -- Calm down -- Clearing the dust -- Proof -- The big bang -- Evolution -- Evil -- Religion -- A crisis of reason.

This is the first comprehensive study of the political philosophy of the British philosopher R. G. Collingwood, best known for his contributions to aesthetics and the philosophy of history. However his political thought, and in particular his book The New Leviathan, have been neglected, even dismissed in some quarters. Professor Boucher argues for the importance of this political theory and provides a perspicuous account of its development and originality. He contends that The New Leviathan is an attempt to reconcile (...) philosophy and history, theory and practice. Collingwood's distinctive contribution to modern political and social thought is seen as his sustained project of distinguishing utility from right, and right from duty; the passion for history coincides with the ethical thought because Collingwood wishes to identify dutiful, or moral, action with a historical civilization. Drawing on a wealth of manuscript material, this book will prove invaluable to political philosophers and intellectual historians. (shrink)