A new reading and comparative interpretation of Gödel’s completeness (1930) and incompleteness (1931) theorems

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself

Toulmin and the Mathematicians: A Radical Extension of the Agenda

Toulmin is famously seen as the progenitor of informal logic and the related theory of argument and is first among many who seek to move the study of argument away from its roots in formal, especially mathematical, logic. Toulmin’s efforts, however, have been substantively criticized by Harvey Siegel, among others, for failing to offer the sort of foundation that, according to Siegel, even Toulmin sees to be required lest the theory of inquiry fall to impotent relativism. What I will attempt to indicate in this paper is, that although Toulmin is correct in rejecting mathematical logic as standardly construed as an adequate theory of argument, and logical empiricist constructions as an adequate basis for the philosophical understanding of science, there is a significant role for metamathematics in the new logic. In particular, I will show how a formal model based on mature physical science rather than arithmetic furnishes crucial support to Toulmin, furnishing philosophical metaphors that afford the foundational support required for normativity and the clarification of key logical concepts required for a robust normative theory of argument in the context of inquiry

The logical anti-psychologism of Frege and Husserl

Frege and Husserl are both recognized for their significant contributions to the overthrowing of logical psychologism, at least in its 19th century forms. Between Frege's profound impact on modern logic that extended the influence of his anti-psychologism and Husserl's extensive attempts at the refutation of logical psychologism in the Prolegomena to Logical Investigations, these arguments are generally understood as successful. This paper attempts to account for the development of these two anti-psychologistic conceptions of logical objects and for some of the basic differences between them. It identifies some problems that are common to strongly anti-psychologistic conceptions of logic and compares the extent to which Frege's and Husserl's views are open to these problems. Accordingly, this paper is divided into two parts. Part I develops a conception of the problems of logical psychologism as they are distinctively understood by each philosopher, out of the explicit arguments and criticisms made against the view in the texts. This conception is in each case informed by the overall historical trajectories of each philosopher's philosophical development. Part II examines the two views in light of common problems of anti-psychologism

Separating Functional Computation from Relations

The logical foundation of arithmetic generally starts with a quantificational logic over relations. Of course, one often wishes to have a formal treatment of functions within this setting. Both Hilbert and Church added choice operators (such as the epsilon operator) to logic in order to coerce relations that happen to encode functions into actual functions. Others have extended the term language with confluent term rewriting in order to encode functional computation as rewriting to a normal form. We take a different approach that does not extend the underlying logic with either choice principles or with an equality theory. Instead, we use the familiar two-phase construction of focused proofs and capture functional computation entirely within one of these phases. As a result, our logic remains purely relational even when it is computing functions

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Poincaré on the Foundation of Geometry in the Understanding

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them