Article of the Week about
Modern mathematics treats space quite differently from classical mathematics.
In ancient mathematics, "space" was a geometric abstraction of the
three-dimensional space observed in the everyday life. The axiomatic method was the main research tool since Euclid (about 300 BC). The coordinate method (analytic geometry) was added by René Descartes in 1637. At that time geometric theorems were treated as an absolute objective truth that can be known through intuition and reason, similar to the objects of natural science; and axioms were treated as obvious implications of definitions.
Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by projective geometry (Gaspard Monge, 1795), corresponds to projective transformations. Not only ellipses but also parabolas and hyperbolas turn into circles under appropriate projective transformations; they all are projectively equivalent figures.
The relation between the two geometries, Euclidean and projective, shows that mathematical objects are not given to us with their structure. Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory.
Distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question "what is the sum of the three angles of a triangle" is meaningful in the Euclidean geometry but meaningless in the projective geometry.
A different situation appeared in the 19th century: (Read more...)
| 
