Hilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p() from [] whether p() has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories (...) IE1 or IU-1. (shrink)

We show that if R is a nonconstant regular (semi-)local subring of a rational function field over an algebraically closed field of characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer; that is, there is no algorithm to decide whether an arbitrary Diophantine equation over R has solutions over R or not. This result can be seen as evidence for the fact that the corresponding problem for the full rational field is also (...) unsolvable. (shrink)

Classical results of additive number theory lead to the undecidability of the existence of solutions for diophantine equations in given special sets of integers. Those sets which are images of polynomials are covered by a more general result in the second section. In contrast, restricting diophantine equations to images of exponential functions with natural bases leads to decidable problems, as proved in the third section.

The paper establishes lower bounds on the provability of.

Stylometry can be defined as the use of numerical methods for the solution of literary problems, most often problems of authorship, integrity, and chronology. As stylometry has been described it seems hardly more than the application of common sense to a literary situation. For example: It consists in collecting as many peculiarities of style and grammar as possible from these works [the dialogues of Plato], particularly the Laws, which are known, or for good reasons supposed to belong to the author's (...) latest period, and observing the frequency with which these occur in other dialogues. If it is then found, e.g., that one dialogue uses commonly 100 of these, another but 60, it is reasonable to suppose the former to be nearer in time to the Laws, i.e. later. (shrink)

Stylometry can be defined as the use of numerical methods for the solution of literary problems, most often problems of authorship, integrity, and chronology. As stylometry has been described it seems hardly more than the application of common sense to a literary situation. For example: It consists in collecting as many peculiarities of style and grammar as possible from these works [the dialogues of Plato], particularly the Laws, which are known, or for good reasons supposed to belong to the author's (...) latest period, and observing the frequency with which these occur in other dialogues. If it is then found, e.g., that one dialogue uses commonly 100 of these, another but 60, it is reasonable to suppose the former to be nearer in time to the Laws, i.e. later. (shrink)

Colour has often been supposed to be a subjective property, a property to be analysed orretly in terms of the phenomenological aspects of human expereince. In contrast with subjectivism, an objectivist analysis of color takes color to be a property objects possess in themselves, independently of the character of human perceptual expereince. David Hilbert defends a form of objectivism that identifies color with a physical property of surfaces - their spectral reflectance. This analysis of color is shown to provide a (...) more adequate account of the features of human color vision than its subjectivist rivals. The author's account of colro also recognises that the human perceptual system provides a limited and idiosyncratic picture of the world. These limitations are shown to be consistent with a realist account of colour and to provide the necessary tools for giving an analysis of common sense knowledge of color phenomena. (shrink)

We recall the characterisation of positive definite polynomial functions over a real closed ring due to Dickmann, and give a new proof of this result, based upon ideas of Abraham Robinson. In addition we isolate the class of convexly ordered valuation rings for which this characterisation holds.

Although it has been something of a fetish for philosophers to distinguish between hallucination and illusion, the enduring problems for philosophy of perception that both phenomena present are not essentially different. Hallucination, in its pure philosophical form, is just another example of the philosopher’s penchant for considering extreme and extremely idealized cases in order to understand the ordinary. The problem that has driven much philosophical thinking about perception is the problem of how to reconcile our evident direct perceptual (...) contact with objects and properties with the equally evident fact that there is no phenomenological signal separating error and truth. “The obscure object of hallucination” offers a subtle and plausible solution to this problem and one that solves the problem generally, not just in the special case of hallucination. Johnston’s objective is to offer a theory of perception that meets two constraints: (1) that it provide an explanation of the possibility of delusive and veridical sensings that are indistinguishable from the first-person perspective and (2) that it count as form of direct realism where this is taken to involve acquaintance with the objects of perception. Johnston uses the first constraint to rule out disjunctivism. The second constraint is used to rule out conjunctivism, which as Johnston uses the term, includes most of the widely adopted philosophical theories of perception. Johnston also develops his own sophisticated and interesting theory of perception. In what follows, I will discuss the relation of Johnston’s theory to conjunctivism, examine one of his anti-conjunctivist arguments and finally compare Johnston’s theory with some other versions of direct realism. These topics constitute a very incomplete selection of the important issues discussed in this rich and interesting paper. I will also not disagree, in any fundamental way, with any of the central theses of Johnston’s discussion.. (shrink)

Herschel's experiments on radiant heat are analysed to see how he understood the role of experiment and how he handled potential difficulties in measurement. He believed that experiments could answer essential questions about nature and was willing to change his mind in light of evidence. Potential problems with data did not shake his confidence in the results of his experiments. Herschel's critic, Leslie, had even less patience with experimental results that did not fit his theory. His harsh condemnations of Herschel's (...) work could not be empirically substantiated, but they pointed to problems of which Herschel was aware. Although the theories of both men about radiant heat have subsequently been abandoned, attempts to address their concerns and differences contributed to the development of the electromagnetic spectrum. The unification of radiant heat and light through the spectrum came about after better instruments made it possible to account for the rival claims of earlier experimenters. (shrink)

We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of ${\mathbb {Q}}$ , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 10.2 (2003) 161-167 [Access article in PDF] Social, Cognitive, and Neural Constraints on Subjectivity and Agency:Implications for Dissociative Identity Disorder Peter Q. Deeley In this commentary, I consider Matthew's argument after making some general observations about dissociative identity disorder (DID). In contrast to Matthew's statement that "cases of DID, although not science fiction, are extraordinary" (p. 148), I believe that there are natural analogs of (...) the disorder that, when considered, make it seem less puzzling and exotic. After discussing these examples, I examine the relations between social, cognitive, and neural processes supporting subjectivity and agency, to identify sources of personal identity in cases of DID that can be used as a basis for assessing Matthew's points against Behnke and Sinnott-Armstrong. I am not proposing a comprehensive theory, but rather identifying what I believe are components of such a theory. I then outline an approach to determining criminal responsibility in DID, drawing on analogies with other psychiatric disorders. Dissociative Identity Disorder DID is a disorder described in DSM-IV, which "is characterised by the presence of two or more distinct identities or personality states that recurrently take control of the individual's behaviour accompanied by an inability to recall important personal information that is too extensive to be explained by ordinary forgetfulness" (American Psychiatric Association, p. 519). Recognizing Personal Identity How is personal identity recognized in normal cognition? If we take personhood to involve recognition of continuity of social, psychological, and physical identity for a given individual, then such judgments are routinely made rapidly and automatically as an aspect of what has come to be termed social cognition (Adolphs 2003). There is likely to be considerable redundancy in [End Page 161] the types of information used to make judgments about whether a person is the same as one who was previously encountered, or perhaps heard about—for example, facial features, movement patterns, speech prosody, dress, speech content, gender, ethnicity, social role, among other features, all provide information about individual identity, and hence personhood. When uncertainty exists, it is likely to reflect a loss or alteration of cues that would ordinarily be used in a rapid, automatic, unexamined way (Adolphs 2003). In such extreme cases (for example, severe mental illness or disfigurement), the problem of identification becomes the subject of explicit reasoning (and hence of potential disagreement, because we then engage in post hoc rationalizations to establish criteria for determining personhood, whereas normally we rely on essentially automatic, implicit abilities that work in most circumstances, some of which make little use of linguistic representations and formal reasoning—for example, individual facial identity recognition, which may occur as quickly as 170 ms after stimulus onset [Adolphs 2003]). Nevertheless, examining our judgments and intuitions in cases that force us to explicitly consider whether personal identity is preserved may provide analogies and clues for how we should proceed with DID. Let us consider, then, some natural analogs of DID. Natural Analogs of DID Method Acting I suggest that method acting is an analog of DID (Manderino 1985). Method acting refers to an approach to acting training and performance in which emphasis is placed on intensely imagining and identifying with the mental life of the character being portrayed. Training courses and manuals list many techniques and practices that can be used to facilitate absorption in a role, drawing on different kinds of memory—for example, experiential memories, memories of others, general knowledge (semantic memory), and memories of sensations and feelings, to creative a vivid sense of the subjective experience and actions of the character. Manderino, a method acting teacher, comments: "it is important to investigate the way in which your own mental makeup can improperly influence a character. Insecurities, self-consciousness, shyness or self-doubts can maintain a strong hold, and permitting any of these feelings to enter a performance is to remain the actor and not enter into the mental life of the character. For some, acting allows personal problems to be left behind, and they are thankful for the complete escape that... (shrink)

For a ring R, Hilbert’s Tenth Problem $HTP$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP$. In contrast, we show that every Turing degree contains a (...) set W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$. (shrink)

Thomas Christiano argues that democracies acquire a right to rule by being the unique embodiment of publicly accessible rules. Justice requires the equal advancement of the interests of all. However, due to the need for citizens to shape a common world despite disagreement and limitations of human cognition, publicity is a necessary constraint on the pursuit of justice. Given that democracy is necessary to secure public equality, democratic authority is thus justified, as democracy is the only political arrangement that satisfies (...) both the publicity constraint and equality requirement. Christiano’s argument depends on a claim that individuals should be given the right to advance what they take to be their interests. This right is defended through two interlocking claims. First, according individuals this right advances three fundamental interests: i) correcting cognitive biases; ii) being at home in the world; and iii) equal moral standing. Second, the three fundamental interests imply that any collective decision-making procedure must be publicly accessible. Thus, any argument against the right to judge for oneself either violates the publicity requirement or the equality constraint. In this paper, I argue that Christiano’s argument faces a problem—under a two-stage plural voting system, an unequal distribution of voting rights can satisfy both the publicity requirement and the equality constraint. Thus, Christiano’s claim that public equality requires an equal distribution of the right to judge for oneself is mistaken. (shrink)

Thomas Christiano argues that democracies acquire a right to rule by being the unique embodiment of publicly accessible rules. Justice requires the equal advancement of the interests of all. However, due to the need for citizens to shape a common world despite disagreement and limitations of human cognition, publicity is a necessary constraint on the pursuit of justice. Given that democracy is necessary to secure public equality, democratic authority is thus justified, as democracy is the only political arrangement that satisfies (...) both the publicity constraint and equality requirement. Christiano’s argument depends on a claim that individuals should be given the right to advance what they take to be their interests. This right is defended through two interlocking claims. First, according individuals this right advances three fundamental interests: i) correcting cognitive biases; ii) being at home in the world; and iii) equal moral standing. Second, the three fundamental interests imply that any collective decision-making procedure must be publicly accessible. Thus, any argument against the right to judge for oneself either violates the publicity requirement or the equality constraint. In this paper, I argue that Christiano’s argument faces a problem—under a two-stage plural voting system, an unequal distribution of voting rights can satisfy both the publicity requirement and the equality constraint. Thus, Christiano’s claim that public equality requires an equal distribution of the right to judge for oneself is mistaken. (shrink)

Revolutions in semiconductor device miniaturization, bioelectronics, and applied neural control technologies are enabling scientists to create machine-assisted minds, science fiction's “cyborgs.” In a paper published in 1999, we sought to draw attention to the advances in prosthetic devices, to the myriad of artificial implants, and to the early developments of this technology in cochlear and retinal implants. Our concern, then and now, was to draw attention to the ethical issues arising from these innovations. Since that time, breakthroughs have occurred at (...) a breathtaking pace. Scientists, researchers, and engineers using differing methodologies are pursuing the possibilities of direct interfaces between brains and machines. Technological innovations as such are neither good nor evil; it is the uses devised for them that create moral implications. As there can be ethical problems inherent in the proper human uses of technologies and because brain chips are a very likely future technology, it is prudent to formulate policies and regulations that will mitigate their ill effects before the technologies are widespread. Unlike genetic technologies, which have received widespread scrutiny within the scientific community, national governments, and international forums, brain–machine interfaces have received little social or ethical scrutiny. However, the potential of this technology to change and significantly affect humans is potentially far greater than that of genetic enhancements, because genetic enhancements are inherently limited by biology and the single location of an individual, whereas hybrids of human and machine are not so restricted. Today, intense interest is focused on the development of drugs to enhance memory; yet, these drugs merely promise an improvement of normal memory, not the encyclopedic recall of a computer-enhanced mind combined with the ability to share information at a distance. The potential of brain chips for transforming humanity are astounding. This paper describes advances in hybrid brain–machine interfaces, offers some likely hypotheses concerning future developments, reflects on the implications of combining cloning and transplanted brain chips, and suggests some potential methods of regulating these technologies. (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

For each $p \geq 2$ there exists a model ${\bf M}^{*}$ of $I\Delta_{0}(\alpha)$ which satisfies the Count( $p$ ) principle. Furthermore, if $p$ contains all prime factors of $q$ there exist $n,r \in {\bf M}^{*}$ and a bijective map $f \in {\rm dom}({\bf M}^{*})$ mapping $\{1,2,\ldots,n\}$ onto $\{1,2,\ldo ts,n+q^{r}\}$ . A corollary is a complete classification of the Count( $q$ ) versus Count( $p$ ) problem. Another corollary shows that the pigeon-hole principle for injective maps does not follow from (...) any of the Count( $q$ ) principles. This solves an open question [Ajtai 94]. (shrink)

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean “small”, (...) in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. (shrink)

We provide a suitable theoretical foundation for the notion of the quantum coherent state which describes the electrostatic field due to a static external macroscopic charge distribution introduced by the author in 1998 and use it to rederive the formulae obtained in 1998 for the inner product of a pair of such states. (We also correct an incorrect factor of 4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\pi$$\end{document} in some of those formulae.) Contrary to what one might expect, (...) this inner product is usually non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves are different. We actually display two different frameworks that lead to the same inner-product formulae, in the second of which Gauss’s law only holds in expectation value. We propose an experiment capable of ruling out the latter framework. We then address the problem of finding a product picture for QED—i.e. a reformulation in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a Hilbert space which is a subspace (the physical subspace) of the full tensor product of a charged-matter Hilbert space and an electromagnetic-field Hilbert space. (The traditional Coulomb gauge formulation of QED isn’t a product picture in this sense because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) Motivated by the first framework for our coherent-state construction, we find such a product picture and exhibit its equivalence with Coulomb gauge QED both for a charged Dirac field and also for a system of non-relativistic charged balls. For each of these systems, in all states in the physical subspace (including the vacuum in the case of the Dirac field) the charged matter is entangled with longitudinal photons and Gauss’s law holds as an operator equation; albeit the electric field operator (and therefore also the full Hamiltonian) while self-adjoint on the physical subspace, fails to be self-adjoint on the full tensor-product Hilbert space. The inner products of our electrostatic coherent states and the product picture for QED are relevant as analogues to quantities that play a rôle in the author’s matter-gravity entanglement hypothesis. Also, the product picture provides a temporal gauge quantization of QED which appears to be free from the difficulties which plagued previous approaches to temporal-gauge quantization. (shrink)

Neil D. Jones, Computability and Complexity. From a Programming Perspective.Neil D. Jones, T. AE. Mogensen, Computability by Functional Languages.M. H. Sorensen, Hilbert's Tenth Problem.A. M. Ben-Amram, The Existence of Optimal Algorithms.

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim (...) is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

Conventional and well-established guidelines for the ethical conduct of clinical research are necessary but not sufficient for addressing research dilemmas related to public health research. There is a particular need for a public health ethics framework when, in the face of an epidemic, research is urgently needed to promote the common good. While there is limited experience in the use of a public health ethics framework, the value and potential of such an approach is increasingly being appreciated. Here we use (...) two examples of adolescent women as potential candidates for participation in microbicide trials to illustrate how ethical decisions for public health research can be enhanced by drawing on both traditional research ethics guidance, and the emerging framework for public health ethics. (shrink)

In a famous lecture in 1900, David Hilbert listed 23 difficult problems he felt deserved the attention of mathematicians in the coming century. His conviction of the solvability of every mathematical problem was a powerful incentive to future generations: ``Wir müssen wissen. Wir werden wissen.'' (We must know. We will know.) Some of these problems were solved quickly, others might never be completed, but all have influenced mathematics. Later, Hilbert highlighted the need to clarify the methods of mathematical reasoning, (...) using a formal system of explicit assumptions, or axioms. Hilbert's vision was the culmination of 2,000 years of mathematics going back to Euclidean geometry. He stipulated that such a formal axiomatic system should be both `consistent' (free of contradictions) and `complete' (in that it represents all the truth). Hilbert also argued that any wellposed mathematical problem should be `decidable', in the sense that there exists a mechanical procedure, a computer program, for deciding whether something is true or not. Of course, the only problem with this inspiring project is that it turned out to be impossible. (shrink)

Richard Browring called "Isi monogatari" one of the most important and at the same time one of the most mysterious texts of Japanese culture. Created at the beginning of the tenth century, at a turning point, at the dawn of the early and mature phases of Hayan's days, this work was one of the first written in classical Japanese, one of the first works of the genre of monogatari, the first work of the genre Uta-monogatari, and others like that. (...) He then laid down standards for the entire development of the Japanese literary tradition. However, we are not interested in the philological aspect of the problem, and such an aspect that was still on the periphery of the studios "Isi monogatari", namely, the religious and philosophical interpretation of the work. A vivid example of such an interpretation is one of the most ancient commentary on this work, "Isa monogatari zuiṇo:". Next, for convenience, we will call it just "Zuino". (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Notes and Fragments RAILING AGAINST REALISM: PHILOSOPHY AND TO THE LIGHTHOUSE by S. P. ROSENBAUM The argument of Graheim Parkes's "Imagining Reality in To the Lighthouse" is described by its author as "a railing against the realist position" (p. 35) as he understands it primarily in my article "The Philosophical Reedism ofVirginia Woolf." ' Apart from the question of whemer railing is a useful way of conducting an inquiry (...) into literature and philosophy, Parkes's discussion raises several important issues about these connections. The way diey are raised, however, involves various mistedces and muddles that make a brief reply to his eurticle difficult. To begin with a small reveeding example, Parkes asserts that I miss the irony in Virginia Woolfs use of die letters P, Q, and R to describe Mr. Ramsay's problems of diought (p. 41). Here is what I wrote: "By representing Mr. Ramsay 's thinking as proceeding in die very genered form of conventional logical symbols, Virginia Woolf is wittily extending the standard letters used to symbolize die conditional type of argument, If P men Cf (p. 339, with italics added this time). Parkes's misreading of my text does not inspire confidence when he turns to Virginia Woolfs fiction or G. E. Moore's philosophy. On a more significant issue, Parkes complains that I boggle his mind by saying diat philosophically, Moore influenced Virginia Woolf more man anyone else, whereas Platonic ideas abound in her books (p. 35). Of course they do, but not very much in her epistemology, which was the subject of my eurticle.2 What Peirkes does not mention is that Plato was also a profound influence on Moore's philosophy. One of die reasons Moore's influence on Bloomsbury was so strong is that he reinterpreted Plato for them. In her images and ideas Virginia Woolf was influenced direcdy dirough her reading of Plato etnd also indirectly through the work of Plotinus and Moore. These influences are not easily sepeurated, which is why studies of Platonic emd Neoplatonic influences in her work that ignore Moore's significance are so incomplete. 89 90Philosophy and Literature Peirkes claims mat die evidence of Moore's influence on Virginia Woolf is slight (p. 34), and then he concerns himself with the putative influences of German philosophers for whom mere is no evidence at all. In the fifteen years since I wrote my article, the biographical evidence for Moore's influence has continued to accumulate as Virginia Woolfs letters, memoirs, emd dieiries were published. From her first novel, in which Principia Etnica is quoted emd briefly discussed, down to die recently discovered revised version of her last memoir, in which she reaffirms her deep respect for Moore,3 the indications of Moore's significance for Virginia Woolf ought to convince any disinterested reader. It cannot be dismissed with a sneer about "the voguish fascination of the Bloomsbury Group..." (p. 34), especially from someone proposing to study yet again the depth psychology of her fiction via Nietzsche and Heidegger (P- 44). Nothing I say here, of course, will persuade Parkes of the value of philosophical realism for understanding To the Lighthouse because diat philosophy, as he conceives of it in his article, consists of straw. To tedce just one exeimple, Peirkes appeeurs to believe Moore's epistemology etrgues "diat things and persons are what they are independently of human consciousness" (p. 37). There is no space here to explain philosophical realism again to Parkes, odier than to note that it is not inconsistent with the veirious concerns (to use Peurkes's terminology) about "transpersonal imagination" or "participation in phenomena" that he finds so interesting in the novel. But what about die novel? For readers indifferent to whether Moore or Russell, Nietzsche or Heidegger are to be found in To the Lighthouse, the relevant question is what such philosophical interpretations medce of it. To them I ceui only say, reread the novel and then ask yourselves if an account of it diat says nothing about die value of art and almost nodiing about the value oflove bears much resemblance to your reading experience. Parkes thinks the portrait of Mr. Ramsay is unsympathetic because he is criticized emd mocked... (shrink)

It is shown that the application of the Lax-Phillips scattering theory to quantum mechanics provides a natural framework for the realization of the ideas of the “Many-Hilbert-Space” theory of Machida and Namiki to describe the development of decoherence in the process of measurement. We show that if the quantum mechanical evolution is pointwise in time, then decoherence occurs only if the Hamiltonian is time-dependent. If the evolution is not pointwise in time (as in Liouville space), then the decoherence may occur (...) even for closed systems. These conclusions apply as well to the general problem of mixing of states. (shrink)

Quasi-set theory $\cal Q$ allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that $x = x$ is not a formula, if $x$ is an arbitrary term. $\cal Q$ was partially motivated by the problem of non-individuality in quantum mechanics. In this paper I discuss the range of explanatory power of $\cal Q$ for quantum phenomena which demand some notion of indistinguishability among quantum objects. My main focus (...) is on the double-slit experiment, a major physical phenomenon which was never modeled from a quasi-set-theoretic point of view. The double-slit experiment strongly motivates the concept of degrees of indistinguishability within a field-theoretic approach, and that notion is simply missing in $\cal Q$. Nevertheless, other physical situations may eventually demand a revision on quasi-set theory axioms, if someone intends to use it in the quantum realm for the purpose of a clear discussion about non-individuality. I use this opportunity to suggest another way to cope with identity in quantum theories. (shrink)

The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that (...) the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems. (shrink)

In early 1927, Pascual Jordan published his version of what came to be known as the Dirac-Jordan statistical transformation theory. Later that year and partly in response to Jordan, John von Neumann published the modern Hilbert space formalism of quantum mechanics. Central to both formalisms are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems in the new (...) theory. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canonically conjugate variables, i.e., p’s and q’s. Jordan ran into serious difficulties when he tried to extend his approach from quantities with fully continuous spectra to those with wholly or partly discrete spectra. For von Neumann, not constrained by the analogy to classical physics and aware of the daunting mathematical difficulties facing the approach of Jordan ), the solution of a problem in the new quantum mechanics required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Related to their disagreement about the appropriate general formalism for the new theory, Jordan and von Neumann stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan was the first to state those rules in full generality, von Neumann rephrased them and then sought to derive them from more basic considerations. In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and Nordheim. We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics. (shrink)

We show that the D A -unification problem is undecidable. That is, given two binary function symbols $\bigoplus$ and $\bigotimes$ , variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following D A -axioms hold: \begin{align*}(x \bigoplus y) \bigotimes z &= (x \bigotimes z) \bigoplus (y \bigotimes z),\\x \bigotimes (y \bigoplus z) &= (x \bigotimes y) \bigoplus (x \bigotimes z),\\x \bigoplus (y \bigoplus z) &= (x \bigoplus y) \bigoplus z.\end{align*} Two (...) terms are D A -unifiable (i.e. an equation is solvable in D A ) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory D A . This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained. (shrink)

Since my article in C.Q. xli, pp. 86 ff., a further discussion of the problem has come to my notice. H. Jacobsohn, in an article entitled Σκνθικ in Zeitschr. f. vergleichende Sprachforschung, liv, pp. 254 ff., anticipates my point that the Greek ᾊξενƿς is borrowed not from Avestan but from some other Iranian language, probably Scythian. He also makes outan attractive case, based on the word παφδεισ¿ς, for considering the Iranian pronunciation at the period when the loan occurred to (...) have been αχšēna, with ē from original ai. It is unfortunate that he should have supported his case with the argument that a form αχšaina is too remote from Greek ξειν¿ς for the identification to have taken place: this insistence on complete homonymy in folk-etymology I have already criticized in discussing Moorhouse's article; and in any case it is by no means certain that ξεινႿς is the earliest form of the name in Greek. (shrink)

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator (...) conjugate to the number operator. (shrink)

We show that the $\mathrm{D_A}$-unification problem is undecidable. That is, given two binary function symbols $\bigoplus$ and $\bigotimes$, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following $\mathrm{D_A}$-axioms hold: \begin{align*}(x \bigoplus y) \bigotimes z &= (x \bigotimes z) \bigoplus (y \bigotimes z),\\x \bigotimes (y \bigoplus z) &= (x \bigotimes y) \bigoplus (x \bigotimes z),\\x \bigoplus (y \bigoplus z) &= (x \bigoplus y) \bigoplus z.\end{align*} Two terms are $\mathrm{D_A}$-unifiable (i.e. an (...) equation is solvable in $\mathrm{D_A}$) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory $\mathrm{D_A}$. This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained. (shrink)