Inverse Problems

Abstract

In this chapter we give illustrations of abductive reasoning that Peirce called “retroduction”. His own paradigm example, the inference to the existence of Napoleon Bonaparte from present documents and monuments, can be generalized to a method of evaluating historical hypotheses by means of the causal traces of past events (Sect. 4.1). Peirce’s thesis about perception as an extreme case of retroduction needs special attention. We argue 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 (Sect. 4.2). 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 (Sect. 4.3). It is further argued that in biology the reconstruction of the evolutionary tree of life on the basis of present evidence is a retroductive task in Peirce’s sense (Sect. 4.4). Starting from the 1860s, the idea of evolution was applied in the study of culture (philology, ethnology, anthropology, folkloristics), with the aim of reconstructing family trees of languages and stemmas of oral and written texts (Sect. 4.5). The inferential abductive structure of textual criticism and stemmatology in cultural sciences is shown to be similar to cladistics in biological phylogenetics.