To found or not to found: that is the question

Aim of this paper is to confute two views, the first about Schr\"oder's presumptive foundationalism, according to he founded mathematics on the calculus of relatives; the second one mantaining that Schr\"oder only in his last years (from 1890 onwards) focused on an universal and symbolic language (by him called pasigraphy). We will argue that, on the one hand Schr\"oder considered the problem of founding mathematics already solved by Dedekind, limiting himself in a mere translation of the Chain Theory in the language of the relatives. On the other hand, we will show that Schr\"oder's pasigraphy was connaturate to himself and that it roots in his very childhood and in his love for foreign languages.Comment: Next to be published in Logic and Logical Philosoph

Binary Relations as a Foundation of Mathematics

We describe a theory for binary relations in the Zermelo-Fraenkel style. We choose for ZFCU, a variant of ZFC Set theory in which the Axiom of Foundation is replaced by an axiom allowing for non-wellfounded sets. The theory of binary relations is shown to be equi-consistent ZFCU by constructing a model for the theory of binary relations in ZFU and vice versa. Thus, binary relations are a foundation for mathematics in the same sense as sets are

The utterly prosaic connection between physics and mathematics

Eugene Wigner famously argued for the "unreasonable effectiveness of mathematics" for describing physics and other natural sciences in his 1960 essay. That essay has now led to some 55 years of (sometimes anguished) soul searching --- responses range from "So what? Why do you think we developed mathematics in the first place?", through to extremely speculative ruminations on the existence of the universe (multiverse) as a purely mathematical entity --- the Mathematical Universe Hypothesis. In the current essay I will steer an utterly prosaic middle course: Much of the mathematics we develop is informed by physics questions we are tying to solve; and those physics questions for which the most utilitarian mathematics has successfully been developed are typically those where the best physics progress has been made.Comment: 12 pages. Minor edits on an essay written for the 2015 FQXi essay contest: "Trick or truth: The mysterious connection between physics and mathematics

Elementary quotient completion

We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that the elementary quotient completion can be obtained as the composite of two free constructions: one adds effective quotients, and the other forces extensionality of maps. We also prove that each construction preserves comprehensions

A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte