AXDEF

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. 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.

The main objectives will be 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.

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 dating back at least to modern age;

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.

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. Difficulties in the investigations on the topic of definitions are their ubiquity, ambiguity and technicality in axiomatic presentations, which explains their lack of a systematic treatment. These difficulties will be addressed and resolved through the creation of a broad consortium of philosophers, logicians, and mathematicians with expertise in the study of the history of science but also in the more formal developments of contemporary logic; the collaborative expertise acquired through previous joint researches between the French principal investigator and the other scientific partners; the use of collaborative work platforms; the presentation and discussion of research products with international colleagues at scientific workshops and seminars.

Expected results include 1) the appraisal of the role of definitions in the development of hypotethico-deductive axiomatics, 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 nderstanding of the relationship between definitions and deduction, and 5) an analysis of the epistemic roles played by definitions in formal and informal provability. The added value of the project consists in the interdisciplinary joint work of experts in different domains, as well as in an effort of offering a first systematic historical and theoretical treatment of axiomatic definitions. The project will contribute new knowledge on axiomatic definitions, both on the role they played in the history of axiomatics and on the epistemic roles they play in contemporary philosophy.