@incollection {381663, title = {Abduction as a strategy for concept formation in mathematics: {C}ardano postulating a negative.}, booktitle = {Abduction and the process of scientific discovery}, year = {2007}, pages = {179{\textendash}194}, publisher = {Centro de Filosofia das Ci{\^e}ncias da Universidade de Lisboa}, abstract = {

When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce’s original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics.

}, author = {Heeffer, Albrecht}, editor = {Pombo, Olga and Gerner, Alexander} } @incollection {J:ala, title = {Adaptive Logics for Abduction and the Explication of Explanation-Seeking Processes}, booktitle = {Abduction and the Process of Scientific Discovery}, year = {2007}, pages = {97{\textendash}119}, publisher = {Centro de Filosofia das Ci{\^e}ncias da U. de Lisboa}, address = {Lisboa}, abstract = {

In this paper, I illustrate the main characteristics of abductive reasoning processes by means of an example from the history of the sciences. The example is taken from the history of chemistry and concerns a very small episode from Lavoisier’s struggle with the ‘air’ obtained from mercury oxide. Eventually, this struggle would lead to the discovery of oxygen. I also show that Lavoisier’s reasoning process can be explicated by means of a particular formal logic, namely the adaptive logic LAr. An important property of LAr is that it not only nicely integrates deductive and abductive steps, but that it moreover has a decent proof theory. This proof theory is dynamic, but warrants that the conclusions derived at a given stage are justified in view of the insight in the premises at that stage. Another advantage of the presented logic is that, as compared to other existing systems for abductive reasoning, it is very close to natural reasoning.

}, author = {Meheus, Joke}, editor = {Pombo, Olga and Gerner, Alexander} }