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Axis History and Philosophy of Mathematics

Research Group “Axioms & definitions”


The Research Group aims to investigate the history of the systems of principles in mathematics and their underlying epistemologies. We work on the development of the systems of axioms in arithmetic and geometry from antiquity to the modern age, the changing meaning of principles throughout the centuries, the evolution of the relations between definitions and axioms. Further topics are the historical transformation of the criteria to formulate and accept a definition or an axiom, and the transformation of the definitions and axioms themselves with the progressive extension of the axiomatic method to other disciplines. The array of authors involved ranges from Euclid and Archimedes to early modern mathematicians such as Leibniz and Lambert, to the authors of new axiomatizations of projective geometry, algebra and topology (to mention a few disciplines) and the great epistemological revolutions in the axiomatic method by Pasch, Frege or Hilbert. We will also discuss a few 20th-Century developments, and the epistemology of principles in the current philosophical and mathematical debate.

Coordination: Vincenzo de Risi (CNRS, SPHERE), Paola Cantù (Centre G.-G. Granger)

Session of Tuesday November 6, 2018: 4p.m. – 6p.m., University Paris Diderot, Room 371 Klein*.

  • Georg Schiemer (University of Vienna)
    What are implicit definitions?
    The notion of implicit definition is a fundamental concept in modern mathematics, in particular in formal axiomatics. It is also ubiquitous in contemporary philosophical debates, ranging from the fields of philosophy of science and mathematics to philosophy of logic and language. Despite the relevance of this notion, it is commonly used in an informal way in these debates. The aim of this paper is to make more precise what implicit definitions actually are. We will address this issue by distinguishing between several types of definitions which are usually associated with the expression “implicit definition”. In particular, the focus here will be on a kind of definition given in structural or formal axiomatics, which in general terms is usually understood as providing a definition of the primitives terms of a given axiomatic theory. A central claim in the talk is that it is somewhat unclear how such definitions should be understood semantically. More precisely, we propose that both in modern mathematics and in subsequent philosophical debates one can distinguish between two types of understanding structural definitions semantically, namely, (i) as definitions of the meaning of the primitive terms of a theory and (ii) as definitions of types or concepts of mathematical objects or, alternatively, as their structures. The central aim in the following will be to survey these two conceptions of structural definitions both in the history of mathematics and in selected philosophical debates. The present work presents joint research with Eduardo Giovannini (CONICET Argentina).
  • Discussion on future activities

Session of Thursday April 11, 2019, at Centre G.-G. Granger, Aix-Marseille University
Program tba


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