Remarks on Mathematical Matter and the Role of Mathematical Notions

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