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O'Shea, James
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O'Shea, James
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O'Shea, James
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Now showing 1 - 10 of 13
- PublicationBounds on the levels of composition algebrasCertain families of quaternion and octonion algebras are conjectured to be of level and sublevel n. A proof of this conjecture is offered in the case where n is a power of two. Hoffmann's proof of the existence of infinitely many new values for the level of a quaternion algebra is generalised and adapted. Alternative constructions of quaternion and octonion algebras are introduced and justified in the case where n is a multiple of a two power.
379 - PublicationKant's Theoretical Philosophy: The 'Analytic' TraditionIn a previous article (O’Shea 2006) I provided a concise overview of the reception of Kant’s philosophy among analytic philosophers during the periods from the ‘early analytic’ reactions to Kant in Frege, Russell, Carnap and others, to the systematic Kant-inspired works in epistemology and metaphysics of C. I. Lewis and P. F. Strawson, in particular. In this chapter I use the recently reinvigorated work of Wilfrid Sellars (1912-1989) in the second half of the twentieth century as the basis for presenting some of the most familiar ‘analytic Kantian’ themes that continue to animate current debates. I also argue that the complex relationships between Sellars’ philosophy and Kant’s thought are often misunderstood. Overall the chapter examines Sellars’ analytic appropriations of Kant in three topic-areas of significant current philosophical debate: (1) conceptual analysis and the structure of human knowledge; (2) laws of nature, the causal modalities, and the pragmatic or relative a priori; and (3) the disputes concerning Kant and nonconceptual content.
57 - PublicationComments on Robert Brandom's 'From Empiricism to Expressivism: Brandom Reads Sellars'I am in agreement with most of what is contained in this powerful book. In particular, I find Brandom’s extended defense of his "Kant-Sellars thesis about modality," which is elaborated throughout the central chapters 3 to 6, the constructive heart of the book, to be both innovative and yet interpretively compelling with respect to the real semantic heart of Sellars’ philosophy. I myself agree that one of the deepest of insights elaborated in different ways by Kant, C. I. Lewis, Sellars, and now Brandom is that the objective purport of any empirical concept presupposes the prescription of lawful modal constraints governing the objects themselves, as systematically reflected in our inferential practices. This is Kant’s idea of the categories as concepts that prescribe laws to appearances, and Sellars’ idea of concepts as involving laws, and C.I. Lewis defends this, too-and now I think that Bob has really developed that in a nice way. And I also agree that these sort of normative presuppositional relationships, in general, constitute a key strand running throughout Sellars’ philosophy: Sellars’ idea, for example, that the normative espousal of principles is reflected in uniformities of practice and in certain natural regularities. (I tried to make that the center of my 2007 book on Sellars as well, in not nearly as sophisticated a way, however.) This highlights the pervasive Janus-faced interplay between what is explicitly asserted on the one hand, and the various normative practices and natural regularities that are thereby, Sellars will say, presupposed or conveyed or prescribed on the other.
62 - PublicationSums of squares in certain quaternion and octonion algebrasFormulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented.
430Scopus© Citations 2 - PublicationLevels and sublevels of composition algebrasLewis' and Leep's bounds on the level and sublevel of quaternion algebras are extended to the class of composition algebras. Some simple constructions of composition algebras of known level values are given. In addition, octonion algebras of sublevel 3 are presented.
284Scopus© Citations 4 - PublicationLevels and sublevels of composition algebras over p-adic function fieldsIn [O'S], the level and sublevel of composition algebras are studied, wherein these quantities are determined for those algebras defined over local fields. In this paper, the level and sublevel of composition algebras, of dimension 4 and 8 over rational function fields over local non-dyadic fields, are determined completely in terms of the local ramification data of the algebras. The proofs are based on the "classification" of quadratic forms over such fields, as is given in [PS].
213Scopus© Citations 1 - PublicationWhat to Take Away from Sellars’s Kantian NaturalismI contend that Sellars defends a uniquely Kantian naturalist outlook both in general and more particularly in relation to the nature and status of what he calls ‘epistemic principles’; and I attempt to show that this remains a plausible and distinctive position even when detached from Sellars’s quasi-Kantian transcendental idealist contention that the perceptible objects of the manifest image strictly speaking do not exist, i.e., as conceived within that common sense framework. I first explain the complex Kant-inspired sense in which Sellars did not take the latter thesis concerning the objects of the manifest image to apply, at least in certain fundamental respects, to persons. In this primary Kantian sense, I suggest, persons as thinkers and agents exist univocally across both the manifest and scientific images, and this in principle would enable an integration of persons within a multi-leveled naturalistic ontology, one that is independent of Sellars’s quasi-Kantian transcendental idealist thesis. Finally, I examine in some detail how this defensible blend of Kantian and naturalist themes turns out to be what is fundamental in Sellars’s complex and controversial views on the nature and status of epistemic principles.
120 - PublicationConcepts of Objects as Involving Laws: A Kantian and Pragmatist Line of ThoughtThis paper traces a Kantian and pragmatist line of thinking that connects the ideas of conceptual content, object cognition, and modal constraints in the form of counterfactual sustaining causal laws. It is an idea that extends from Kant’s Critique of Pure Reason through C. I. Lewis’s Mind and the World-Order to the Kantian naturalism of Wilfrid Sellars and the analytic pragmatism of Robert Brandom. Kant put forward what I characterize as a modal conception of objectivity, which he developed as an extended argument stretching from the transcendental deduction through the analogies of experience to the regulative maxims of reason and reflective judgment. In related ways in Lewis and Sellars, the very idea of an object of knowledge (and of intentionality more generally) is connected with a certain lawfulness or modal constraint the necessary representation of which, they argue, is an achievement of conceptualization. While Sellars agreed with the spirit of Lewis’s famous pragmatic conception of the a priori, Sellars’s conception of meaning and conceptual content differed in crucial ways with important consequences for this issue. I argue furthermore that a certain phenomenalist temptation threatens to spoil this insight both among some of Kant’s interpreters and in Lewis’s thought. Finally, I point out that Brandom’s "Kant-Sellars thesis" provides new support for this line of thought. Although questions concerning idealism continue to raise controversies for neo-Kantians and pragmatists, the line of thought itself represents a distinctive and still promising approach to questions concerning intentionality and conceptual content.
96 - PublicationThe weak isotropy of quadratic forms over field extensionsThe weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved.
316Scopus© Citations 1 - PublicationIsotropy over function fields of Pfister formsThe question of which quadratic forms become isotropic when extended to the function field of a given form is studied. A formula for the minimum dimension of the minimal isotropic forms associated to such extensions is given, and some consequences thereof are outlined. Especial attention is devoted to function fields of Pfister forms. Here, the relationship between excellence concepts and the isotropy question is explored. Moreover, in the case where the ground field is formally real and has finite Hasse number, the isotropy question is answered for forms of sufficiently large dimension.
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