Albrecht Heeffer

Using Invariances in Geometrical Diagrams: Della Porta, Kepler and Descartes on Refraction

In this paper, I will demonstrate how geometrical diagrams on refraction were instrumental in the discovery of the sine law of refraction. In particular, I will show how a specific diagram in the Paralipomena assisted Kepler in looking for invariances of proportions under different angles of incidence. Eventually, Kepler failed in finding a quantitative law of refraction, but it will be shown that his basic hypothesis and methodology can lead to the discovery of a quantitative law and that probably this was Descartes’ path to the discovery of the sine law. Both Kepler and Descartes could build on a tradition of geometrical reasoning which accounted for co-exact properties in geometrical diagrams. Della Porta was the first to recognize such properties in diagrams dealing with refraction.

Data-Driven Induction in Scientific Discovery: A Critical Assessment Based on Kepler’s Discoveries

Motivated by the renewed interest in knowledge discovery from data (KDD) by the artificial intelligence community, this paper provides a critical assessment of the model of data-driven induction for scientific discovery. The most influential research program using this model is developed by the BACON team. Two of the main claims by this research program, the descriptive and constructive power of data-driven induction, are evaluated by means of two historical cases studies: the discovery of the sine law of refraction in optics and Kepler’s third law of planetary motion. I will provide evidence that the data used by the BACON program—despite the claims being made—does not correspond with the historical data available to Kepler and his contemporaries. Secondly, it is shown that for the two cases the method by which the general law was arrived at did not involve data-driven induction. Finally, the value of the data-driven induction as a general model for scientific discovery is being questioned.

Handling Inconsistencies in the Early Calculus

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism (C&P) proposed by Brown and Priest in (Journal of Philosophical Logic, 33(4), 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from.

A difficult case : Pacioli and Cardano on the Chinese Rings

Abstract The Chinese rings puzzle is one of those recreational mathematical problems known for several centuries in the West as well as in Asia. Its origin is diffcult to ascertain but is most likely not Chinese. In this paper we provide an English translation, based on a mathematical analysis of the puzzle, of two sixteenth-century witness accounts. The first is by Luca Pacioli and was previously unpublished. The second is by Girolamo Cardano for which we provide an interpretation considerably different from existing translations. Finally, both treatments of the puzzle are compared, pointing out the presence of an implicit idea of non-numerical recursive algorithms.

On Remembering Cardano Anew

Many of the stories surrounding the famous sixteenth century feud between Girolamo Cardano and Niccolo Tartaglia are completely unsupported by the historical record. In particular, stories that have Tartaglia aiding and abetting Cardanos arrest for heresy are chronologically impossible.