Abstract
Every proof is faced with the requirement of proving that the proof is correct, and the proof of the correctness of the proof again meets the same requirement and the proof of the correctness of the correctness of the proof also, etc. In order to escape from an infinite regress into which one is led one has to come down with a purely algorithmic criterion for correctness or to claim that thinking is identical with its subject matter. Whence the preference of number and more generally of conceptualism in pure mathematics. Conceptualism is a kind of nominalism that does not give a realist understanding of mathematics (note that Platonism is not an opponent of nominalism as some seem to believe). The paper presents some examples and reflections intending to hint at the role of formal thought in the process of knowledge growth. It argues that there is no division of labor according to which certain modes of human cognition are associated with certain tasks and certain cognitive roles exclusively. In this connection, the paper claims that the subject matter of mathematical activity is represented within the system of activity by many different means. Mathematics differs in fact from logic in as much as a principle of heterogeneity or of flexible ‘means-objects-relationships’ is valid. Formalization in contrast brings forward a principle of homogeneity — that like follows like. Every subject matter requires principles homogeneous with itself. The paper tries to draw some conclusions from this difference with respect to the role of formalization within human cognitive development.
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Otte, M. Mathematical knowledge and the problem of proof. Educ Stud Math 26, 299–321 (1994). https://doi.org/10.1007/BF01279518
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DOI: https://doi.org/10.1007/BF01279518