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11
PART III
Speak on the topic
1. Read and translate the text
Topic 1
A modern view of geometry
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 "sub-
conscious 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. We call
this laboratory phase in the development of geometry "scientific ge-
ometry". About 600 B.C. the Greeks began to inject deduction into ge-
ometry giving rise to what we call "demonstrative geometry".
In time demonstrative geometry becomes a material-axiomatic
study of idealized physical space and of the shapes, sizes, and relations
of idealized 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 an-
other. 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 axiomatics. There developed the idea of branch of
mathematics as an abstract body of theorems deduced from a set of
12
postulates. Each geometry became, from this point of view, a particular
branch of mathematics.
In the twentieth century the study of abstract spaces was inaugu-
rated 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 ob-
jects are involved, and a 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 un-
derstanding, and the presentation of some concept or development. The
best way to describe geometry today is not as some separate and pre-
scribed 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.
2. Retell the text
PART III postulates. Each geometry became, from this point of view, a particular
branch of mathematics.
Speak on the topic In the twentieth century the study of abstract spaces was inaugu-
rated and some very general studies came into being. A space became
1. Read and translate the text
merely a set of objects together with a set of relations in which the ob-
Topic 1
jects are involved, and a geometry became the theory of such a space.
A modern view of geometry The boundary lines between geometry and other areas of mathematics
became very blurred, if not entirely obliterated.
For a long time geometry was intimately tied to physical space, There are many areas of mathematics where the introduction of
actually beginning as a gradual accumulation of subconscious notions geometrical terminology and procedure greatly simplifies both the un-
about physical space and about forms, content, and spatial relations of derstanding, and the presentation of some concept or development. The
specific objects in that space. We call this very early geometry "sub- best way to describe geometry today is not as some separate and pre-
conscious geometry". Later, human intelligence evolved to the point scribed body of knowledge but as a point of view – a particular way of
where it became possible to consolidate some of the early geometrical looking at a subject. Not only is the language of geometry often much
notions into a collection of somewhat general laws or rules. We call simpler and more elegant than the language of algebra and analysis, but
this laboratory phase in the development of geometry "scientific ge- it is at times possible to carry through rigorous trains of reasoning in
ometry". About 600 B.C. the Greeks began to inject deduction into ge- geometrical terms without translating them into algebra or analysis. A
ometry giving rise to what we call "demonstrative geometry". great deal of modern analysis becomes singularly compact and unified
In time demonstrative geometry becomes a material-axiomatic through the employment of geometrical language and imagery.
study of idealized physical space and of the shapes, sizes, and relations
of idealized physical objects in that space. The Greeks had only one 2. Retell the text
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 an-
other. 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 axiomatics. There developed the idea of branch of
mathematics as an abstract body of theorems deduced from a set of
11 12
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