On Forms of Justification in Set Theory

In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how ‘intuitively plausible’ an axiomis, whereasextrinsicjustification supports an axiom by identifying certain‘desirable’ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation

Anti-Foundational Categorical Structuralism

The aim of this dissertation is to outline and defend the view here dubbed “anti-foundational categorical structuralism” (henceforth AFCS). The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be “the science of structure” expressed in the language of category theory, which is argued to accurately capture the notion of a “structural property”. In characterizing mathematical theorems as both conditional and schematic in form, the program is forced to give up claims to securing the truth of its theorems, as well as give up a semantics which involves reference to special, distinguished “mathematical objects”, or which involves quantification over a fixed domain of such objects. One who wishes—contrary to the AFCS view—to inject mathematics with a “standard” semantics, and to provide a secure epistemic foundation for the theorems of mathematics, in short, one who wishes for a foundation for mathematics, will surely find this view lacking. However, I argue that a satisfactory development of the structuralist view, couched in the language of category theory, accurately represents our best understanding of the content of mathematical theorems and thereby obviates the need for any foundational program

On Forms of Justification in Set Theory

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

BEHAVIORISM AND LOGICAL POSITIVISM: A REVISED ACCOUNT OF THE ALLIANCE (VOLUMES I AND II)

The primary aim of this work is to show that the widespread belief that the major behaviorists drew importantly upon logical positivist philosophy of science in formulating their approach to psychology is ill-founded. Detailed historical analysis of the work of the neobehaviorists Edward C. Tolman, Clark L. Hull, and B. F. Skinner leads to the following conclusions: (1) each did have significant contact with proponents of logical positivism; but (2) their sympathies with logical positivism were quite limited and were restricted to those aspects of logical positivism which they had already arrived at independently; (3) the methods which they are alleged to have imported from logical positivism were actually derived from their own indigenous conceptions of knowledge; and (4) each major neobehaviorist developed and embraced a behavioral epistemology which, far from resting on logical positivist assumptions, actually conflicted squarely with the anti-psychologism that was a cornerstone of logical positivism. It is suggested that the myth of an alliance between behaviorism and logical positivism arose from the incautious interpretations of philosophical reconstructions as historical conclusions. This and other historiographical issues are discussed in the concluding chapter, where it is argued that the anti-psychologism of the logical positivists is an unnecessary impediment to a fuller understanding of the phenomenon of knowledge