The relationship between axioms and definitions has been a key turning point in the contemporary transformation of logic, mathematics and science, following the birth and development of modern axiomatics and the radical changes in the definitions of the fundamental notions of mathematics: number, size, set. It is often said that definitions play no role in axiomatic systems except as abbreviations. Axioms are often considered as implicit definitions or characterizations of the primitive terms that appear in axioms. The project approaches the study of axiomatic definitions from two complementary perspectives: historical and theoretical. Accordingly, the project will have two main objectives, respectively 1) to investigate the origin of the notion of axiomatic definitions, and 2) to classify axiomatic definitions and study the kind of problems they are applied to in logic and philosophy of mathematics.
Research hypotheses
1. The history of axiomatic definitions, their relation to deduction and adequacy criteria. The historical inquiry is based on the general research hypothesis that definitions, together with a larger spectrum of factors, contributed to the transformation of axiomatics into an investigation of hypothetico-deductive systems. This hypothesis will be tested by verifying three more specific claims: 1) the belief that axioms serve to define or characterize mathematical entities did not arise abruptly in the 19th century, but was the result of complex debates about the distinction between axioms and definitions and on the role of deduction; 2) there is no unique sense in which axioms can be considered as implicit definitions in the debates by 19th and 20th century mathematicians and logicians; 3) several criteria for the adequacy of definitions have been discussed in the history of axiomatics, and there is no one theory that has gathered general consensus.
2. The classification and the epistemic role of axiomatic definitions. The theoretical inquiry is based on the general research hypothesis that a more rigorous understanding of the role played by axiomatic definitions would highly contribute to a better understanding of several contemporary debates in philosophy of logic, mathematics and in the philosophy of science. Two specific research hypotheses will guide this workpackage: 1) definitions might play both an inferential and a semantic role; 2) axiomatic definitions are philosophically crucial to understand the contemporary debate on formal and informal provability.
Expected results
1) the appraisal of the role of definitions in the development of hypotethico-deductive axiomatic
2) a detailed history of the relation between axioms and definitions in logic and in mathematics from the Renaissance to early 20th century
3) a classification of definitions occurring in axiomatic presentations
4) a better understanding of the relationship between definitions and deduction
5) an analysis of the epistemic roles played by definitions in formal and informal probability.
Deliverables
1) A monography on the history of axiomatic definitions codirected by Paola Cantù, Vincenzo De Risi and Georg Schiemer to be submitted to the series Frontiers in History of Science. The volume will offer a first comprehensive view on definitions, as well as new insights on the roles and perspectives of deduction and axiomatic.
2) A wide range of scientific articles to be published in international peer-reviewed journals discussing specific historical issues, alternative classifications of axiomatic definitions and how the axioms might have further epistemic functions beside their inferential role.
3) A doctoral dissertation on Bernard Bolzano’s theory of axioms and definitions.
4) A doctoral dissertation on a taxonomy of different kinds of definitions and adequacy criteria in 20th century philosophy of logic and mathematics.
