The vivacity of mathematics results (partly) from the fact that mathematics is stretched between several poles, not being committed to any one. The paper presents the following polarities: realism - idealism, the finite - the infinite, the discrete - the continuous, the approximate - the exact, certitute - probability, simplicity - complexity, unity - multiplicity.
The origins of mathematics, a close connection and interpenetration of its parts, and uniform procedures of dealing with the mathematical matter - all of them speak in favour of the integrality of mathematics. It seems that a strong argument for such a view is a fundamental object of contemporary mathematics; namely a real line, which contains real numbers (so arithmetics as well) and constitutes a basis of geometry, mathematical analysis and all derivative branches. From the basic-structures perspective it is clear (...) that the real line is an exceptionally complex structure, for it contains the ordered structure (generated by the less-than relation), the algebraic structure (generated by addition and multiplication), the gemetrical structure (generated by translations and reflections) and the topological structure (generated by open intervals). This example explains, at least to some extent, the integrality and also the vivacity of mathematics. On the other hand, the integrality has not been confirmed by comprising the whole mathematics in one axiomatised, deductive theory. Moreover, the increasing „volume” of mathematics, unwillingness of mathematicians to cross the specialisation-barriers, emphasis on utility (models) and difficulties with axiomatisations of some parts of mathematics cast some doubts on the integrality in question. (shrink)
Primary object of interest of mathematicians can be identified as a „mathematical matter”, the concept analogous to „physical matter” or „biological matter”. The „mathematical matter” is the soil upon which mathematics grows. One can distinguish three levels of it: some abstract but not necessarily clear conceptions, operational notions (like number) but not necessarily openly defined, theories not necessarily axiomatic. The „mathematical matter” originates in the abstract reflection upon events and forms in time and space. Its important elements are notions formed (...) mostly in the process of idealization and/or abstraction. Once formed, notions usually evolve being, e.g., simplified or complexified. Mathematics is a mirror of the world, most abstract and therefore fundamental. Although it is a free activity of human mind, mathematics reflects some deep ideas (beauty, simplicity etc.) while most interesting notions come up in tension fields between pairs of poles: the notion of a function can be seen as a bridge connecting variability and immutability, while that of a limit – connecting finiteness and infinity. A fusion of freedom and internal restrictions leads to mathematics which is simple, beautiful and effective. (shrink)
The article recalls shortly the early story of cooperation between the already existing Lvov philosophical school, headed by Twardowski, and the just then establishing Warsaw mathematical school, headed by Sierpiski. After that recollection the article proceeds to contributions made by men influenced by the two schools. Most prominent of them was Alfred Tarski whose work in those times, concentrated mainly upon the theory of deduction, axiom of choice, cardinal arithmetic, and measure problem, has been described in some detail.