Euclidean and non Euclidean Geometries
One of Euclid's fundamental contributions to Mathematics, was the concept
of axiomatization. He understood the need for assuming
a number of rather obvious and easily acceptable statements and derive everything else on the basis of those statements. These
statements are what we call axioms. Euclid introduced the axioms for classical geometry and his approach was so groundbreaking, that
classical geometry came to be known as Euclidean geometry.
However, not everything in his program went as smoothly as he would have hoped. While choosing his axioms, he realized that there was one axiom that did not bare the simplicity of the other axioms. He hopelessly tried to find a way to bypass it, but in the end he had to assume its truth and use it as a building block for his theory. This axiom, that became known as "Euclid's 5th postulate" asserted that, given a point outside a straight line, there exists exactly one line that goes through that point and is parallel to the initial line. Just draw a picture and you will realize that this is a very reasonable statement. However it is not as elementary as its siblings and that was what made Euclid reluctant to accept it so easily.
However, not everything in his program went as smoothly as he would have hoped. While choosing his axioms, he realized that there was one axiom that did not bare the simplicity of the other axioms. He hopelessly tried to find a way to bypass it, but in the end he had to assume its truth and use it as a building block for his theory. This axiom, that became known as "Euclid's 5th postulate" asserted that, given a point outside a straight line, there exists exactly one line that goes through that point and is parallel to the initial line. Just draw a picture and you will realize that this is a very reasonable statement. However it is not as elementary as its siblings and that was what made Euclid reluctant to accept it so easily.
His contemporaries seemed to have shared his doubts and started searching for a way to derive (to prove) the 5th postulate from the other
axioms. The search went on for about two thousand years! No satisfactory way to bypass that axiom had been found. It seemed that mathematicians
were getting more and more convinced that the 5th postulate was an essential axiom. But how could they establish this?
The solution came in the most unexpected way. Mathematicians started exploring what would happen if they dared to imagine a universe where the 5th postulate did not hold, a universe that is different than what we see around us. Bolyai, Lobachewsky and Gauss were among the first to envision geometries where either parallel lines did not exist at all or that there were plenty of parallel lines going through that point in the description of the 5th postulate. These geometries ended up being as consistent as Euclidean geometry because all of them could be coordinatized and expressed through numbers. So one has only to believe in numbers and the existence of exotic universes is as probable as the existence of what we perceive as our Euclidean environment.
The solution came in the most unexpected way. Mathematicians started exploring what would happen if they dared to imagine a universe where the 5th postulate did not hold, a universe that is different than what we see around us. Bolyai, Lobachewsky and Gauss were among the first to envision geometries where either parallel lines did not exist at all or that there were plenty of parallel lines going through that point in the description of the 5th postulate. These geometries ended up being as consistent as Euclidean geometry because all of them could be coordinatized and expressed through numbers. So one has only to believe in numbers and the existence of exotic universes is as probable as the existence of what we perceive as our Euclidean environment.