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Ajdukiewicz's account of mathematical theories is presented and analyzed. Theories consist of primary (original) and secondary (derivative) theorems. Theories go through three phases or stages: (a) preaxiomatic and intuitive, (b) axiomatic but intuitive, (c) axiomatic and abstract, whereas the final stage takes two forms: definitional and formal. Each stage is analyzed. The role of the concepts of truth, evidence, consequence, and existence is examined. It is claimed that the second stage is apparent or transitory, whereas the initial and final stages are vital and constitute two salient attitudes to mathematics, focused on truth or consequence respectively. It is also claimed they are attitudes rather than stages, and the crucial difference between them concerns effectiveness. The chief question of philosophy of mathematics turns out to be to determine whether mathematical theories are assertive or hypothetical.

DOI: 10.1007/s11212-016-9245-x

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