Proof: Some notes on a phenomenon between freedom and enforcement Gregor Nickel * Fachbereich Mathematik, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Ger- many E-mail: nickel@mathematik.uni-siegen.de 1 Introductory remarks and a disclaimer Proof lies at the core of mathematics. A large part of every day’s work in mathematical research is devoted to searching for provable theorems and their proofs, explaining proofs to students or colleagues, and reading other mathematician’s proofs and trying to understand them. Mathematics is special among the other sciences due to the role of proof and insofar as mathematical proof offers the highest possible rigor. Moreover, it is a spe- cial feature of mathematics—in contrast to all positive sciences—that it can work on its foundations using its own methods. The mathematical analysis of mathematical proof is systematically developed in 20th century’s formal logic and (mathematical) proof theory. 1 Here it was possible to transform proof from the process of mathematical arguing to an object—e.g., a well- formed sequence of symbols of some formal language—mathematics can argue about. Though this approach offers valuable results I will not con- centrate on it. In fact, there has been a growing interest in ‘non-formal’ features of mathematics. In fact, a closer look at the ‘real existing’ mathematics ex- hibits a much less formal science than the formalistic picture claimed. Here historical studies, 2 arguments from ‘working mathematicians’, 3 and socio- logical studies 4 point into a similar direction. Concerning a non-formal ‘phe- nomenology of proof’ we refer to the recent discussion initiated by Yehuda * The author would like to thank two anonymous referees for their prompt reports containing instructive and often helpful criticism. 1 In his comprehensive work on the philosophy of mathematical proof theory, Wille (2008) emphasizes that proof theory is special among the mathematical subdisciplines because it discusses normative questions about external justifications of its axiomatic basis. In fact, it can not be denied that the content of the publications in proof theory vary between a purely mathematical and a less technical one comprising normative claims and argumentations. However, in every mathematical area there are normative discussions— however rarely published—about the ‘sense’ of axioms, definitions, the fruitfulness of results etc. The only difference seems to be the communication medium. 2 Cf., e.g., the pioneering work of Imre Lakatos (1976). 3 Cf., e.g., the polemic in (Davis and Hersh, 1981) against a philosophy of mathematics emphasizing only the formal aspects. 4 Cf., e.g., (Heintz, 2000). Benedikt L¨ owe, Thomas M¨ uller (eds.). PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications, London, 2010. Texts in Philosophy 11; pp. 281–291. Received by the editors: 16 December 2009; 4 February 2010; 16 February 2010. Accepted for publication: 22 February 2010.