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In his booklet ‘Contributions to a better founded presentation of mathematics’ of 1810 Bernard Bolzano made his first serious attempt to explain the notion of a rigorous proof. Although the system of logic he employed at that stage is in various respects far below the level of the achievements in his later Wissenschaftslehre, there is a striking continuity between his earlier and later work as regards the methodological constraints on rigorous proofs. This paper tries to give a perspicuous and critical account of the fragmentary logic of Beyträge, and it shows that there is a tension between that logic and Bolzano's methodological ban on ‘kind crossing’. [ABSTRACT FROM PUBLISHER]

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet ‘Contributions to a better founded presentation of mathematics’ of 1810 and to his ‘Theory of science’ (1837). I shall try to defend the following contentions: (1) Bolzanos’ vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion differs from his stance to indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. (3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. [ABSTRACT FROM PUBLISHER]