Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics

Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics

Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics

Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics

Excerpt

It is the aim of these reflections to give an insight into the nature of mathematical concept formation, that is, to point out in the activities of the mathematician what might be of interest to a philosophically minded observer. This immediately differentiates the volume from textbooks of mathematics. The reader will not find here a system of theorems with completely developed proofs. He will find no calculations of examples nor applications of mathematics. All that is pushed into the background in favor of a presentation of mathematical ideas.

In the first place, we shall be concerned here in greater detail with the structure of the realm of numbers. The choice of this topic needs a brief justification. Proceeding from intuitive points of view, Leibniz and Newton created differential and integral calculus. In the eighteenth century these investigations soared extraordinarily, one brilliant discovery following another in the sphere of pure analysis as well as in the domain of their applications. This period of mathematics has been compared, not unjustly, with the period of the great discoverers and the heroes of the sea. The mathematicians of that age had the feeling of stepping into a new intellectual world, eager to explore the contours of the continent that sprang up before them out of the mist. In odd contrast to this series of wonderful discoveries was the obscurity that spread over the foundations of the entire concept creation. It cannot be maintained that Leibniz and Newton were very clear about the meaning of a . . .

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