Nominalism In Mathematics - Modality And Naturalism

I defend modal nominalism in philosophy of mathematics - under which quantification over mathematical ontology is replaced with various modal assertions - against two sources of resistance: that modal nominalists face difficulties justifying the modal assertions that figure in their theories, and that modal nominalism is incompatible with mathematical naturalism. Shapiro argues that modal nominalists invoke primitive modal concepts and that they are thereby unable to justify the various modal assertions that figure in their theories. The platonist, meanwhile, can appeal to the set-theoretic reduction of modality, and so can justify assertions about what is logically possible through an appeal to what exists in the set-theoretic hierarchy. In chapter one, I illustrate the modal involvement of the major modal nominalist views (Chihara\u27s Constructibility Theory, Field\u27s fictionalism, and Hellman\u27s Modal Structuralism). Chapter two provides an analysis of Shapiro\u27s criticism, and a partial response to it. A response is provided in full in chapter three, in which I argue that reducing modality does not provide a means for justifying modal assertions, vitiating the accusation that modal nominalists are particularly burdened by their inability to justify modal assertions. Chapter four discusses Burgess\u27s naturalistic objection that nominalism is unscientific. I argue that Burgess\u27s naturalism is inadequately resourced to expose nominalism (modal or otherwise) as unscientific in a way that would compel a naturalist to reject nominalism. I also argue that Burgess\u27s favored moderate platonism is also guilty of being unscientific. Chapter five discusses some objections derived from Maddy\u27s naturalism, one according to which modal nominalism fails to affirm or support mathematical method, and a second according to which modal nominalism fails to be contained or accommodated by mathematical method. Though both objections serve as evidence that modal nominalism is incompatible with Maddy\u27s naturalism, I argue that Maddy\u27s naturalism is implausibly strong and that modal nominalism is compatible with forms of naturalism that relax the stronger of Maddy\u27s naturalistic principles

Russell’s method of analysis and the axioms of mathematics

In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s method of analysis, which are intended to shed light on his view about the status of mathematical axioms. I describe the position Russell develops in consequence as “immanent logicism,” in contrast to what Irving (1989) describes as “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament, and to propose a method for discovering structural relationships of dependence within mathematical theories

Theoretical analysis of the philosophy and practice of disciplined inquiry

2015 Spring.Includes bibliographical references.This dissertation theoretically examined the process of disciplined inquiry in the social sciences from its philosophical foundations to its extensions into practice. Key to conceptualization of disciplined inquiry were two regulative ideals: the commitment to the concepts that define the possibility of experience and the commitment to processes for combining the concepts of experience. The paradigm theory of Lincoln, Lynham, and Guba (e.g., Lincoln & Lynham, 2011; Lincoln, Lynham, & Guba, 2011) provided a sophisticated explanation of the possibility of experience that inquirers can commit to when engaging in disciplined inquires. Review of literature revealed an inadequacy in the state of theoretical understanding of processes for combining the concepts of experience. To develop a theoretical agenda of research for disciplined inquiry, the literature on paradigm theory and theory building was analyzed. A historical analysis of paradigm theory revealed milestones in more than 40 years of inquiry focused on conceptualization of the theory. A reverse engineering analysis theoretically examined paradigm theory and its milestones identified from the historical analysis for key features of the theoretical process. A revised conceptualization of disciplined inquiry was presented and a theoretical agenda for developing the underlying theoretical framework for the processes of combining the concepts of experience was outlined

From Political Liberalism to Para-Liberalism: Epistemological Pluralism, Cognitive Liberalism & Authentic Choice

Advocates of political liberalism hold it as a superior alternative to perfectionism on the grounds that it avoids superfluous and/or controversial claims in favor of a maximally-inclusive approach undergirded by a free-standing justification for the ideology. These assertions prove difficult to defend: political interpretations of liberalism tend to be implicitly ethnocentric; they often rely upon a number of controversial, and even empirically falsified, assumptions about rationality--and in many ways prove more parochial than their perfectionist cousins. It is possible to reform political liberalism to address these challenges, but generally at the expense of the supposed normative force and universality of the liberal project. However, this para-liberal approach is much better in keeping with contemporary findings in sociology, psychology and cognitive science--and can much more effectively accommodate the illiberal challenge

Atheism and Agatheism in the Global Ethical Discourse: Reply to Millican and Thornhill-Miller

Peter Millican and Branden Thornhill-Miller have recently argued that contradictions between different religious belief systems, in conjunction with the host of defeaters based on empirical research concerning alleged sources of evidence for ‘perceived supernatural agency’, render all ‘first-order’, that is actual, religious traditions positively irrational, and a source of discord on a global scale. However, since the authors recognise that the ‘secularisation thesis’ appears to be incorrect, and that empirical research provides evidence that religious belief also has beneficial individual and social effects, they put forward a hypothesis of a ‘second-order religious belief ’, with Universalist overtones and thus free of intergroup conflict, and free of irrationality, since supported (solely) by the Fine-Tuning Argument. While granting most of their arguments based on empirical research and embracing the new paradigm of the atheism/religion debate implicit in their paper, I contend that Millican’s and Thornhill-Miller’s proposal is unlikely to appeal to religious believers, because it misconstrues the nature and grounds of religious belief. I suggest that their hypothesis may be refined by taking into account a view of axiologically grounded religious belief that I refer to as ‘agatheism’, since it identifies God or the Ultimate Reality with the ultimate good (to agathon)