Understanding Mathematics Essays

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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
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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 psychophysical creations, logical structures sensu largo etc.

2) it consists in grasping relations which determine the sense of what is to be understood.

3) it enables the reconstruction and the application, in other conditions, of what we understand.

The essence of this kind of cognition, which we call understanding, decides about the hypothetical
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