Philosophical Foundations of Mathematics

Source abstract

Currently, the deductive-axiomatic method of forming mathematical knowledge is the most widespread. It allows for compactness, logical closure, rigor, and consistency in the presentation of any established mathematical theory. However, the aforementioned advantages of this method are counterbalanced by its significant drawbacks, stemming from an overload of logical issues and formal details. The limitations of the logical deductive method of constructing mathematics can be surmounted by adopting a novel, ennoiative (derived from the Greek ένvοια [ennoia] – meaning, concept) approach to comprehending it. It is based on the empirically confirmed premise that the laws governing reality are laws of reason, inherently mathematical in nature. Its implementation necessitates a multi-level grasp of mathematics, allowing for the study of its epistemological origins, understanding the genesis of mathematical ideas and the internal logic of mathematical methods and theories, mastering a set of mathematical algorithms, and acquiring problem-solving methods. This leads to perceiving mathematics as the most perfect method of cognition. Only such an approach to presenting mathematics allows for its genuine comprehension. The article presents the philosophical foundations of the new, ennoiative approach to understanding and forming mathematical knowledge, proposed and implemented by the author.