Geometry, Reasoning and Imagination
The monumental work of Euclid “Elements” definitely laid a kind of a foundation for the study of geometry (and Mathematics in general) But Euclid would not have imagined the plane E^2 or the space E^3 as a set of ordered pairs and ordered triples. The notion of cartesian (rectangular) coordinates was put forth only in the 17th century by Descarte (and independantly by Fermat).
We shall see the contributions of Euclid to Mathematics and then discuss some contemperory issues. Euclid and his disciples came up with an axiomatic set-up to describe basic geometric attributes like lines, planes, triangles, points etc. Thus certain basic definitions were made and a few axioms were made that were intuitively acceptable. Theorems and propositions were proved starting from the defined terms and axioms. For instance high school geometry teaches us that the sum of angles of a triangle(drawn on a plane) equals to two right angles(180 degrees).
I take a digression here to emphasize on a certain development of Mathematical thinking: Euclid has shown us the fertile world of Imagination both geometrically as well as logically. Here I shall dwell upon the logical part. The property of triangles alluded to earlier is “deduced” by certain properties like congruences observed when a line is drawn transversely to a pair of parallel straight lines. This aspect of deduction is all pervading through mathematics. Let us see some general examples of inference. Suppose aristotle observes on several occassions that birds are generally hatched through eggs. The scientific reasoning of the philosopher aristotle would make him believe that all birds come up by way of eggs. But this statement is rendered false the moment he observes some kind of a bird that is born in another way. (I do not know if such a bird exists,..this is just for argument sake). This kind of inference from a frequent observation is called Inductive logic and the reasoning established in this way is only probabilistically measured. The deductive logic on the other hand is stronger in the sense that as long as the basic axioms are true the deduced property/phenomena has to be true. For example If Virendra Sehwag is batting in the last two balls of the last over and the score reads 275.... and if we know that the final score is 283, no sixer was hit and no extras were given(including overthrow runs), then the only possibility we deduce is that the two balls went for two boundaries!!!. [THE BIRD EXAMPLE IS FROM The Math Explorer-by J.H.Weaver Universities Press]
Euclid's work did not involve coordinates as he did not need them. However the technological advances from 15th century onwards especially Newtonian Mechanics required precise measurement for instantaneous calculations. As far as Euclid's geometry (Note that what Mathematicians call “Euclidean geometry” is different) is concerned, mundane calculations like area of closed curves, surfaces and volumes of certain surfaces like pyramids etc all of them could be done easily. However the thoughts of the renaissance era required study of dynamics involving time, like the motion of stars, predicting the time of eclipses etc. Hence the gift of Cartesian geometry became indispensable. At this point I quote Peter Doyle (of the University of Illinois, Urbana Champaign) “The spirit of Mathematics is not captured by spending 3 hours of solving 20 look alike homework problems. Mathematics is thinking, comparing, analyzing, inventing and understanding. The main point is not quantity or speed, the main point is quality of thought....reach a more complete and a better understanding...”
Marriage between Algebra and geometry: The introduction of cartesian coordinates lead to a sort of marriage between algebra and geometry. Geometry on one hand uses the spatial sense of the brain say while thinking about navigation, painting symmetrical designs , thinking aesthetically for interior designs and architecture etc. On the other hand proving theorems in geometry requires the verbal function of the brain where one analysies logically the properties that ought to be satisfied by the shapes in consideration. Remarkably geometers invented a way to look at geometry without ever drawing any figure. For instance we may imagine a person chewing gum and then assume that we can establish a relation between the amount of gum he/she chews and the linear distance travelled by him. Let us say this relation is given by the equation y^2=5x-x^3. Now irrespective of the geometric shape of this function, one still analyses the relation.
This is true as the geometer in this case analyses only the functions involved theirein.
Analytic geometry is a more appropriate word since we have several coordinates other than cartesian coordinates like the spherical ones, polar coordinates etc. In fact an attempt to understand geometry in a coordinate free form lead to generalizations like Riemannian geometry. Here Mathematicians wanted to freely (and smoothly) traverse between one coordinate system to another. To conclude this piece, we can look at other generalizations like the plane and space (R^2 and R^3) will take us further into an n-dimensional euclidean space. One can even go on to replace the real number system by the complex numbers to obtain the space C^n. Algebraic geometers use appropriate rings in place of R and C(These are also rings !) to define affine spaces on which geometry is done.
