This book examines the philosophical conception of abductive reasoning as developed by Charles S. Peirce, the founder of American pragmatism. It explores the historical and systematic connections of Peirce's original ideas and debates about their interpretations. Abduction is understood in a broad sense which covers the discovery and pursuit of hypotheses and inference to the best explanation. The analysis presents fresh insights into this notion of reasoning, which derives from effects to causes or from surprising observations to explanatory theories. The (...) author outlines some logical and AI approaches to abduction as well as studies various kinds of inverse problems in astronomy, physics, medicine, biology, and human sciences to provide examples of retroductions and abductions. The discussion covers also everyday examples with the implication of this notion in detective stories, one of Peirce’s own favorite themes. The author uses Bayesian probabilities to argue that explanatory abduction is a method of confirmation. He uses his own account of truth approximation to reformulate abduction as inference which leads to the truthlikeness of its conclusion. This allows a powerful abductive defense of scientific realism. This up-to-date survey and defense of the Peircean view of abduction may very well help researchers, students, and philosophers better understand the logic of truth-seeking. (shrink)

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are typically allowed only (...) with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ?given?, that is, known or ?data?, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ?unknown? objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky. (shrink)

Charles S. Peirce introduced in the late 19th century the notion of abduction as inference from effects to causes, or from observational data to explanatory theories. Abductive reasoning has become a major theme in contemporary logic, philosophy of science, and artificial intelligence. This paper argues that the new growing branch of applied mathematics called inverse problems deals successfully with various kinds of abductive inference within a variety of scientific disciplines. The fundamental theorem about the inverse reconstruction of plane functions from (...) their line integrals was proved by Johann Radon already in 1917. The practical applications of Radon’s theorem and its generalizations include computerized tomography which became a routine imaging technique of diagnostic medicine in the 1970s.Keywords: Abduction; Inference to the best explanation; Inverse problems; Peirce; Radon’s theorem; Tomography. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...) a theoretical model suitable for describing the tran-sition from the conjecturing to the proving phase. In the paper such a model is introduced and worked out to test Lakatos' machinery of proofs and refutations from a new point of view. Abduction and its categorical counter-part, adjunction, allow to explain within a unifying framework most of the phenomenology of conjectures and proofs, encompassing also the method of Greek analysis-synthesis. (shrink)

Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.