Mathematical quantum theory I: Random ultrafilters as hidden variables

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables.More precisely, these constructions permit us to write down a category-theoretically natural correlation between ideal outcomes of quantum measurements u of a universal wave function, and possible worlds of an Everett-Wheeler-like many-worlds-theory.The universal wave function, first, is simply a pure state of the Hilbert space (L ₂([0, 1]) ^M in a model M an appropriate mathematical-physical theory T, where T includes enough set-theory to derive all the analysis needed for von Neumann-algebraic formulations of quantum theory.The worlds of this framework can then be given a genuine model-theoretic construal: they are random models M(u) determined by M-random elements u of the unit interval [0, 1], where M is again a fixed model of T.Each choice of a fixed basis for a Hilbert space H in a model of M of T then assigns ideal spectral values for observables A on H (random ultrafilters on the range of A regarded as a projection-valued measure) to such M-random reals u. If is the universal Lebesgue measure-algebra on [0, 1], these assignments are interrelated by the spectral functional calculus with value 1 in the boolean extension (V( )) ^M , and therefore in each M(u).Finally, each such M-random u also generates a corresponding extension M(u) of M, in which ideal outcomes of measurements of all observables A in states are determined by the assignments just mentioned from the random spectral values u for the universal position-observable on L ₂([0, 1]) in M.At the suggestion of the essay's referee, I plan to draw on its ideas in the projected sequel to examine more recent modal and decoherence-interpretations of quantum theory, as well as Schrödinger's traditional construal of time-evolution. A preliminary account of the latter — an obvious prerequisite for any serious many-worlds-theory, given that Everett's original intention was to integrate time-evolution and wave-function collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measure-algebras to reinterpret time-ordered processes of measurements (determined, for example, by a given Hamiltonian observable H in M) as individual measurements in somewhat more complexly defined extensions M(u) of M.In plainer English: if one takes a little care to distinguish boolean- from measure-algebraic tensor-products of the universal measure-algebra L, one can reinterpret formal time-evolution so that it becomes internal to the universal random models M(u).     

Natural deduction rules for English

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as the c, a (an) c, some c, any c, and every c, where c represents a common noun. These latter expressions are called quantifiers, and other expressions of the form that c or that c itself, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that Every man loves some woman is implied by, but does not imply, Some woman is loved by every man, since the latter is taken to mean the same as Some woman is loved by all men. Also, Jack knows which woman came is implied by Some woman is known by Jack to have come, but not by Jack knows that some woman came.     

The array size function in simple visual search tasks: A comparison between “go-no go” and “yes-no” tasks under conditions of high and low target-noise similarity

Summary In research on visual search within a single eye-fixation a number of different tasks are used and referred to interchangeably. Research with other types of tasks suggests that there are possibly important differences between these tasks. In the present study, two types of search tasks were compared under conditions as equal as possible: the go-no go task and the yes-no task. Conditions of low and high target-noise similarity were used. The results obtained showed: a) a steeper slope of the array size function in yes-no tasks than in go-no go tasks on the first day of practice but not on the second: b) a higher intercept value of the same function for yes-no tasks than for go-no go tasks; and c) a greater proportion of errors with yes-no tasks than with comparable go-no go tasks. A tentative model, describing the main features of the results obtained, is briefly sketched.     

Leadership of antireligious propaganda in the Soviet Union

Re-emergent scientific atheism bears the marks of its historical origins in the efforts of Bon-Bruevi and Jaroslavskij. The disciples of the Lenin generation use their fathers somewhat as second-level classics.     

Can Parts of Space Move? On Paragraph Six of Newton’s Scholium

Paragraph 6 of Newtons Scholium argues that the parts of space cannot move. A premise of the argument – that parts have individuality only through an order of position – has drawn distinguished modern support yet little agreement among interpretations of the paragraph. I argue that the paragraph offers an a priori, metaphysical argument for absolute motion, an argument which is invalid. That order of position is powerless to distinguish one part of Euclidean space from any other has gone virtually unremarked. It remains uncertain what the import of the paragraph is but it is not close to apparently similar arguments of Leibniz.     

Utility theory with inexact preferences and degrees of preference

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.     

Demonstrative utterances

Conclusion We have worked out a small part of a program originally devised by Kaplan which included a clarification of the phrase y intends to demonstrate x. The analysis of demonstrative utterances, combined with a theory of deferred ostensive reference, enabled us to account for a much larger class of utterances (accompained by a relevant gesture) than expected. Regrettably, the analysis forces more questions on us than it answers. Although I have used the term gesture quite freely without providing any sort of explication, there is much more to say about gestures and their relation to speech. Especially questions concerning the ontological nature of gestures (What kind of a thing is a gesture anyway?), their epistemological status (How can we tell gestures apart from other actions?), and their semantic status (What makes a gesture have meaning?) have remained largely unexplored. To my knowledge, no philosopher in the analytic tradition has addressed those issues in any systematic manner.     

Tarski and the Metalinguistic Liar

I offer an interpretation of a familiar, but poorly understood portion of Tarskis work on truth – bringing to light a number of unnoticed aspects of Tarskis work. A serious misreading of this part of Tarski to be found in Scott Soames Understanding Truth is treated in detail. Soamesreading vies with the textual evidence, and would make Tarskis position inconsistent in an unsubtle way. I show that Soames does not finally have a coherent interpretation of Tarski. This is unfortunate, since Soames ultimately arrogates to himself a key position that he has denied to Tarski and which is rightfully Tarskis own.     

Epistemologische betrachtungen zu [S4, S5]

Zusammenfassung Die zahlreichen modallogischen Systeme zwischen den Standardkalkülen S4 und S5 werden vom epistemologischen Standpunkt aus untersucht, indem Notwendigkeit wahlweise als Wissen bzw. als Überzeugt-sein gedeutet wird. Dabei stellt sich heraus, daß — unter gewissen andernorts begründeten Voraussetzungen über epistemische Logik-S4.4 als Logik der wahren Überzeugungen aufgefaßt werden kann, während die Systeme S4.3.2 und S4.2 als Logiken für solche Leute erscheinen, die das Schema Wissen = wahre Überzeugung nur eingeschränkt für ganz spezielle rein doxastische bzw. rein epistemische Sätze akzeptieren. S4.2 ist dabei allem Anschein nach die Logik des Wissens.

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The numerous modal systems between S4 and S5 are investigated from an epistemological point of view by interpreting necessity either as knowledge or as (strong) belief. It is shown that-granted some assumptions about epistemic logic for which the author has argued elsewhere-the system S4.4 may be interpreted as the logic of true belief, while S4.3.2 and S4.2 may be taken to represent epistemic logic systems for individuals who accept the scheme knowledge = true belief only for certain special instances. There is strong evidence in favor of the assumption that S4.2 is the logic of knowledge.

     

The Dancing Philosopher

This excerpt from Kenneth Kings essay, The Dancing Philosopher, traces its genesis from Nietzsches Thus Spoke Zarathustra (a work that greatly impacted Isadora Duncans founding of modern dance) that, in tandem with the emerging technology of the writing machine (typewriter), camera and kinetoscope (cinematography), conjoined the kinetropic and lexigraphemic to inaugurate the kinetic cogito. Maurice Merleau-Pontys phenomenological exposition of corporeality further amplified the reflexive potential of movement and the philosophical understanding of kinesthesia, and King cites as well the technosophic synergy of John Cages and Merce Cunninghams long artistic collaboration that furthered the frontier of a mind-body epistemic.     

Praxis and Agency in Foucault’s Historiography

This paper examines the consequences for agency that Foucaults historiographical approach constructs. The analysis begins by explaining the difference between legislative history and exemplary history, drawing parallels to similar theoretical distinctions offered in the works of Max Weber, J.L. Austin, and Zygmunt Bauman. The analysis continues by reading Habermass critique of Foucault through the tropological lenses suggested by White [Metahistory. Baltimore, MD: Johns Hopkins University Press, 1973]; it argues that Habermass critique misrecognizes the tropes of Foucaultian genealogy. The paper draws implications for education by articulating possibilities for praxis and agency in terms of pedagogy specifically related to the distinction between didactics and modeling. The paper concludes by suggesting that genealogy does not play by Hegels rules, but rather exemplifies agency in ways that are not recognizable from a modernist perspective.     

From negation to “notion”: Cognitive processes and argumentative strategies

Summary This article deals with the role of negation as a language and cognitive operation. Such a topic is treated here within the framework of the argumentative strategies which consist in making certain cognitive landmarks of the discourse flip over with the intent of imposing the necessity to choose between two types of notions, aiming at the transformation of this choice into an implication. The reference here to the Aristotelian logic of Prior Analytics appears to be more efficient than any other contemporary logic and the author intends to give account of the role of negation as contrary coming into play on an operational and cognitive basis in all the argumentative strategies which oscillate reciprocally from universal to particular.     

The history of philosophy in contemporary philosophy: The view from Germany

Conclusion I have frequently mentioned objective problems and topics in the preceding sections. But what exactly is the force of objective here? As my remarks should have made clear I have been using objective to contrast with purely historical. A purely historical approach never gets beyond reproduction, commentary, and interpretation. I call an approach objective when it involves a philosopher who advances his own theses and claims.This minimal understanding of objectivity (in the context of my remarks in this paper) by no means implies that there are problems and topics, systems of concepts, methods, and similar factors that are eternal, completely independent of the contingencies of history (of philosophy, of the sciences), that are not relative to a language, to a logic, to a model, etc. Indeed whether there are problems, etc., in just this absolute, atemporal sense is itself a question for systematic philosophy. It seems clear that the formulation of a problem can only take place against a cognitive background of some sort and within some conceptual scheme.³⁴ Such an assumption is made by most if not all analytic philosophers. But the fact that a philosophical tradition recognizes conceptual schemes does not make it a purely historical, non-objective philosophy, in the sense already introduced and described. A philosopher who explicitly accepts a certain conceptual scheme proceeds in an entirely objective and systematic (and not purely historical) manner when, within this framework, he formulates his own theses.This paper is the text of a talk. the title is due to Barry Smith.     

Contextualism and the Skeptic: Comments on Engel

Mylan Engels paper (2004) is divided into two parts: a negative part, criticizing the costs of contextualism and a constructive part proposing a noncontextualist resolution of the skeptical problem. I will only address the constructive part here. The constructive part is composed of three elements: (i) a reconstruction or reformulation of the original skeptical argument, which draws on the notion of epistemic possibility (e-possibility), (ii) a distinction between two senses of knowledge (and two corresponding kinds of e-possibility): fallibilistic and infallibilistic, and (iii) an argument which tries to hoist the skeptic by their own petard, namely the closure principle (CP). As I will argue, there are two ways to understand Engels anti-skeptical argument. Only in one interpretation does the argument depend on the proposed reconstruction of the skeptical argument in terms of e-possibility. But this version of the argument is unsound. More importantly, the skeptic has a strong prima facie objection at her disposal, which applies to both interpretations of the argument. If this objection is valid, Engels argument does not hold. But once it is invalidated, his argument is superfluous.     

Intentionalism and Intransitivity

I argue in this paper that the existence of sorites series of color patches – series of color patches arranged so that the patches on each end look different in color though no two adjacent patches do – shows that the relation of same phenomenal charac­ter as is not a transitive relation. I then argue that the intransitivity of same phenomenal character as conflicts with certain versions of intentionalism, the view that an experiences phenomenal character is exhausted, or fully determined by its intentional content. Lastly, I consider various objections to the arguments and reply to them.     

Beyond verisimilitude: A linguistically invariant basis for scientific progress

This paper proposes a solution to David Miller's Minnesotan-Arizonan demonstration of the language dependence of truthlikeness (Miller 1974), along with Miller's first-order demonstration of the same (Miller 1978). It is assumed, with Peter Urbach, that the implication of these demonstrations is that the very notion of truthlikeness is intrinsically language dependent and thus non-objective. As such, truthlikeness cannot supply a basis for an objective account of scientific progress. I argue that, while Miller is correct in arguing that the number of true atomic sentences of a false theory is language dependent, the number of known sentences (under certain straightforward assumptions) is conserved by translation; degree of knowledge, unlike truthlikeness, is thus a linguistically invariant notion. It is concluded that the objectivity of scientific progress must be grounded on the fact (noted in Cohen 1980) that knowledge, not mere truth, is the aim of science.For criticism and comments I am indebted to Noretta Koertge, David Miller, and an anonymous Synthese referee.     

Was Marx an ethical humanist?

Zusammenfassung Während der letzten zehn Jahre wurde viel über den Humanismus des jungen Marx gesprochen. Osteuropäische Marxisten, die bemüht sind, ihren Anti-Stalinismus durch Berufung auf die Autorität von Marx selbst zu untermauern, gebrauchen den Ausdruck Humanismus in einem ungenauen Sinn, etwa gleichbedeutend mit Anthropozentrismus. Aber wenn man sagt, daß Marx Haltung anthropozentrisch sei, so sind damit die Hauptfragen erst gestellt, nicht schon gelöst.Humanismus mag etwa soviel wie Säkularismus bedeuten — der Mensch, nicht Gott, wird als im Mittelpunkt stehend gedacht. Die anthropozentrische Haltung kann verschiedene Formen annehmen, vor allem die Formen, die man als Humanismus der Ideale und als Humanismus der Prinzipien bezeichnen könnte. Der Humanismus der Ideale ist ausdrücklich an derZukunft orientiert, der Humanismus der Prinzipien aber an derGegenwart. Nur ein Humanismus der Prinzipien, dem es darum geht, den Eigenwert existierender Individuen zu behaupten und zu verteidigen, verdient es, ethischer Humanismus genannt zu werden. Marx, sogar der jüngste Marx, war kein ethischer Humanist in diesem Sinn. Und nur ein Humanismus in diesem strengen Sinn würde einen philosophischen Standort bieten, von dem aus man den Stalinismus oder Neo-Stalinismus angreifen könnte.Marx war ein Säkularist, und er entwarf ein humanistisches Ideal für die Zukunft, aber humanistische Prinzipien für die Gegenwart lehnte er ab. Er betonte, daß nur dem nicht entfremdeten, produktiven Individuum der kommunistischen Zukunft ein eigener Wert zukomme. Bis dahin haben Individuen nur einen geschichtlich instrumentalen Wert: jene, die an der Verwirklichung der kommunistischen Gesellschaft arbeiten, sind zu respektieren; diejenigen, welche dabei Widerstand leisten oder versagen, müssen ausschließlich als Hindernisse auf dem Wege des geschichtlichen Fortschritts behandelt werden.In diesem Sinn ist der Leninismus und sogar der Stalinismus und Neo-Stalinismus in dem zukunftsorientierten Humanismus der Ideale des jungen Marx einbegriffen, oder zumindest davon nicht ausgeschlossen. Selbst der jüngste Marx machte sich den Modeirrtum des 19. Jahrhunderts zu eigen — den Irrtum des aufgeschobenen Wertes oder des zeitlich verstellten Wertes und nahm damit eine mit dem ethischen Humanismus grundsätzlich unvereinbare Position ein.

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An earlier, and much shorter, version of this paper was read at a session on Marxism and Humanism at the Fourteenth International Philosophy Congress in Vienna, September 4, 1968, and published in Vol. II of the Congress Proceedings, Vienna, 1968, pp. 69–73.     

Finitude and Hume’s Principle

The paper formulates and proves a strengthening of Freges Theorem, which states that axioms for second-order arithmetic are derivable in second-order logic from Humes Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. Finite Humes Principle also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Freges definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.     

Stuffed Cabbage in the Old New School Cafeteria

The purpose of this lecture is to celebrate the memory of Aron Gurwitsch by examining and enlarging the domain of phenomenological clarification of some elements of what Gurwitsch called the logic of reality. Chief among those elements are the nature of the taken-for-grantedness of our existential belief, the difference between presentive and non-presentive indices of reality and the ground for the self-illumination of the world of working.     

A notorious affair called exportation

In Quantifiers and Propositional Attitudes, Quine held (a) that the rule of exportation is always admissible, and (b) that there is a significant distinction between a believes-true (Ex)Fx and (Ex) a believes-true F of x. An argument of Hintikka's, also urged by Sleigh, persuaded him that these two intuitions are incompatible; and he consequently repudiated the rule of exportation. Hintikka and Kaplan propose to restrict exportation and quantifying in to favoured contexts — Hintikka to contexts where the believer knows who or what the person or thing in question is; Kaplan to contexts where the believer possesses a vivid name of the person or thing in question. The bulk of this paper is taken up with criticisms of these proposals. Its ultimate purpose, however, is to motivate an alternative approach, which imposes no restrictions on exportation or quantifying in, but repudiates Quine's other intuition: this is the approach taken in my A Logical Form for the Propositional Attitudes.This paper is based on my doctoral dissertation (Rockefeller University, 1977). I wish to thank Susan Haack for her help in turning a draft into the present version.