Volume 34, 1998
Philosophy of Mathematics
The Problems of Understanding Mathematics
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.