A modern view of geometry

Date: 2015-10-07; view: 892.

Text 12. Make the written translation into Russian (time 90 minutes)

(1790 characters)

For a long time geometry was intimately tied to physical space, actually beginning as a gradual accumulation of subconscious notions about physical space and about forms, content, and spatial relations of specific objects in that space. We call this very early geometry «subconscious geometry». Later, human intelligence evolved to the point where it became possible to consolidate some of the early geometrical notions into a collection of somewhat general laws or rules.

In time demonstrative geometry becomes a study of physical space and of the shapes, sizes, and relations of physical objects in that space. The Greeks had only one space and one geometry; these were absolute concepts. The space was not thought of as a collection of points but rather as a realm or locus, in which objects could be freely moved about and compared with one another. From this point of view, the basic relation in geometry was that of congruence.

With the elaboration of analytic geometry in the first half of the seventeenth century, space came to be regarded as a collection of points; and with the invention, about two hundred years later of the classical non-Euclidean geometries. But space was still regarded as a locus in which figures could be compared with one another. Geometry came to be rather far removed from its former intimate connection with physical space, and it became a relatively simple matter to invent new and even bizarre geometries.

At the end of the last century, Hilbert and others formulated the concept of formal axiomatic. There developed the idea of branch of mathematics as an abstract body of theorems deduced from a set of postulates. Each geometry became, from this point of view, a particular branch of mathematics.

In the twentieth century the study of abstract spaces was inaugurated and some very general studies came into being. A space became merely a set of objects together with a set of relations in which the objects are involved, and geometry became the theory of such a space.

The boundary lines between geometry and other areas of mathematics became very blurred, if not entirely obliterated.

There are many areas of mathematics where the introduction of geometrical terminology and procedure greatly simplifies both the understanding, and the presentation of some concept or development. The best way to describe geometry today is not as some separate and prescribed body of knowledge but as a point of view – a particular way of looking at a subject. Not only is the language of geometry often much simpler and more elegant than the language of algebra and analysis, but it is at times possible to carry through rigorous trains of reasoning in geometrical terms without translating them into algebra or analysis. A great deal of modern analysis becomes singularly compact and unified through the employment of geometrical language and imagery.