Is God a Mathematician_ - Mario Livio [72]
Space is not an empirical concept which has been derived from external experience…Space is a necessary representation a priori, forming the very foundation of all external intuitions…On this necessity of an a priori representation of space rests the apodictic certainty of all geometrical principles, and the possibility of their construction a priori. For if the intuition of space were a concept gained a posteriori, borrowed from general external experience, the first principles of mathematical definition would be nothing but perceptions. They would be exposed to all the accidents of perception, and there being but one straight line between two points would not be a necessity, but only something taught in each case by experience.
To put it simply, according to Kant, if we perceive an object, then necessarily this object is spatial and Euclidean.
Hume’s and Kant’s ideas bring to the forefront the two rather different, but equally important aspects that had been historically associated with Euclidean geometry. The first was the statement that Euclidean geometry represents the only accurate description of physical space. The second was the identification of Euclidean geometry with a firm, decisive, and infallible deductive structure. Taken together, these two presumed properties provided mathematicians, scientists, and philosophers with what they regarded as the strongest evidence that informative, inescapable truths about the universe do exist. Until the nineteenth century these statements were taken for granted. But were they actually true?
The foundations of Euclidean geometry were laid around 300 BC by the Greek mathematician Euclid of Alexandria. In a monumental thirteen-volume opus entitled The Elements, Euclid attempted to erect geometry on a well-defined logical base. He started with ten axioms assumed to be indisputably true and sought to prove a large number of propositions on the basis of those postulates by nothing other than logical deductions.
The first four Euclidean axioms were extremely simple and exquisitely concise. For instance, the first axiom read: “Between any two points a straight line may be drawn.” The fourth one stated: “All right angles are equal.” By contrast, the fifth axiom, known as the “parallel postulate,” was more complicated in its formulation and considerably less self-evident: “If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, then the two lines inevitably will intersect each other if extended sufficiently on that side.” Figure 39 demonstrates graphically the contents of this axiom. While no one doubted the truth of this statement, it lacked the compelling simplicity of the other axioms. All indications are that even Euclid himself was not entirely happy with his fifth postulate—the proofs of the first twenty-eight propositions in The Elements do not make use of it. The equivalent version of the “fifth” most cited today appeared first in commentaries by the Greek mathematician Proclus in the fifth century, but it is generally known as the “Playfair axiom,” after the Scottish mathematician John Playfair (1748–1819). It states: “Given a line and a point not on the line, it is possible to draw exactly one line parallel to the given line through that point” (see figure 40). The two versions of the axiom are equivalent in the sense that Playfair’s axiom (together with the other axioms) necessarily implies Euclid’s original fifth axiom and vice versa.
Figure 39
Over the centuries, the increasing discontent with the fifth axiom resulted in a number of unsuccessful attempts to actually prove it from the other nine axioms or to replace it by a more obvious postulate. When those efforts failed, other geometers began trying to answer an intriguing