Mathematical Determinacy

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing indeterminacy from pluralism and clarifying the crucial notion of open-endedness. It then critically discusses the two major strategies for securing arithmetical categoricity using open-endedness; one based on a collapse theorem, the other on a kind of anti-overspill idea. With this done, a new argument for the categoricity of arithmetic is then presented. In subsequent discussion, the philosophical importance of this categoricity result is called into question to some degree. The categoricity argument is then supplemented by an appeal to the infinitary omega rule, and an argument is given that beings like us can actually follow the omega rule without any violation of Church’s thesis. Finally, the chapter discusses the extension of this type of approach beyond arithmetic, to set theory and the rest of mathematics.

Elements of concrete algorithmics: computability and solvability

An approach to proving the fundamental results of the theory of recursive functions using specific algorithms is consider. For this, the basic constructions of the algorithm are describing exactly and Church's thesis for more narrow classes of algorithmically computational functions is specified (concretized). Using this approach, the belonging of functions to classes of algorithmically computable is argued by the construction of the corresponding algorithms.

Is Church’s Thesis Still Relevant?

The article analyses the role of Church’s Thesis (hereinafter CT) in the context of the development of hypercomputation research. The text begins by presenting various views on the essence of computer science and the limitations of its methods. Then CT and its importance in determining the limits of methods used by computer science is presented. Basing on the above explanations, the work goes on to characterize various proposals of hypercomputation showing their relative power in relation to the arithmetic hierarchy.The general theme of the article is the analysis of mutual relations between the content of CT and the theories of hypercomputation. In the main part of the paper the arguments for abolition of CT caused by the introduction of hypercomputable methods in computer science are presented and critique of these views is presented. The role of the efficiency condition contained in the formulation of CT is stressed. The discussion ends with a summary defending the current status of Church’s thesis within the framework of philosophy and computer science as an important point of reference for determining what the notion of effective calculability really is. The considerations included in this article seem to be quite up-to-date relative to the current state of affairs in computer science.1

What is Computable?

This chapter defines formally the notion of computable function through its inductive and recursive definitions. It covers the construction schemas to define total and partial computable functions. It concludes by explaining how recursive definitions are general and equivalent to other formulations, a result known as Church’s Thesis.

Signal Recovery Beyond Linear Estimates

This chapter explains why Σ‎0 1 formulas of Peano arithmetic (PA) capture all computably enumerable sets, as claimed by Alonzo Church's thesis from the previous chapter. This allows us to capture “computable analysis” in the language of PA, since computable sets and functions are definable in terms of computable enumerability. To justify the claim that Σ‎0 1 = “computably enumerable,” this chapter makes a thorough analysis of the concept of computation. It takes a precise, but intuitively natural, concept of computation and translates it into the language of PA. The chapter demonstrates that the translation is indeed Σ‎0 1, but with a slightly different (though equivalent) definition of Σ‎0 1.