This paper is an attempt to explain the structure of the process of understanding mathematical objects such as notions, definitions, theorems, or mathematical theories. Understanding is an indirect process of cognition which consists in grasping the sense of what is to be understood, showing itself in the ability to apply what is understood in other circumstances and situations. Thus understanding should be treated functionally: as acquiring sense. We can distinguish three basic planes on which the process of understanding mathematics takes place. The first is the plane of understanding the meaning of notions and terms existing in mathematical considerations. A mathematician must have the knowledge of what the given symbols mean and what the corresponding notions denote. On the second plane, understanding concerns the structure of the object of understanding wherein it is the sense of the sequences of the applied notions and terms that is important. The third plane-understanding the 'role' of the object of understanding-consists in fixing the sense of the object of understanding in the context of a greater entity, i.e., it is an investigation of the background of the problem. Additionally, understanding mathematics, to be sufficiently comprehensive, should take into account (apart from the theoretical planes) at least three other connected considerations-historical, methodological and philosophical-as ignoring them results in a superficial and incomplete understanding of mathematics.
In an outstanding book by P. J. Davis and R. Hersh, The Mathematical Experience, there is a small chapter devoted to the crisis of understanding mathematics. Alas, this fragment focuses only on the presentation of the difficulties the students have with understanding a certain proof and some common-sense attempts to explain them, whereas it totally ignores the very notion of understanding. The only statement characterizing this notion is the remark that understanding is connected with effort.
The problem of understanding mathematics requires, in my opinion, a short presentation of a more general issue, that is the issue of understanding as such. I will treat understanding as a kind of indirect cognition, determined by the perception of the relations between the objects of various order (y becomes comprehensible for x as a part of the relation xRy, in which y is an object of a different order intentionally grasped by x). As it can be seen, I neglect here the problem of understanding another human being, although it is usually achieved through understanding the phenomenally accessible human behaviours, i.e. linguistic or extralinguistic creations.
It seems that the Polish philosopher Izydora Dąmbska grasped the problem of understanding accurately and concisely, stating that this kind of cognition is characterized by the following factors:
1) it refers to the objects connected with the spiritual reality-signs, psychic and...