هندسه نااقليدسي ، انقلابي پارادايمي در رياضيات

Non- Eucleadan Geometry, A Paradigmatic Revolution in Mathematics

The picture Kuhn gives for the usual development pattern of nature science can be put in the following manner: Prior facts - normal science crisis --- revolution - modern normal science - modern crisis. Kuhnʹs view is considered to be of significance in that it puts Emphasis on the revolutionary aspect of scientific development (change). Kuhn maintains the scientific change occurs by `revolutionsʹ in which an older paradigm is overthrown and is replaced by a framework incompatible or even incommensurable with it. It is also of importance because it gives a very significant role to the sociological factors (distinctions) in scientific communities. Since the publication of the Structure of Scientific Revolution, there has been the question whether the picture Kuhn gives for the history of natural sciences can be applied to mathematics. It seems the answer is ʹnoʹ. For it is clear that the nature of mathematics is such that it denies the most important aspect or characteristic he attributes to the scientific change, i. e., revolution. In this article, the writer seeks to show that the transition form Eucleadan geometry to the non -Eucleadan geometry of Kuhnʹs revolution lies in mathematics. Of course, this does not mean that all the constituents of what he thinks of developmental pattern of nature science are true of the realm of mathematics, Rather, two important distinctive features, revolutionary and sociological aspects of nature science, can also be applied to the realm of mathematics. In other words, Kuhnʹs idea of `changeʹ is really possible in mathematics. Whenever we are concerned with the Kuhnʹs "paradigm", widely accepted by scientific communities, the change brings about revolution.