… When Euclid stated the fifth postulate, he stood at the parting of the ways. There is nothing comparable to it in the whole history of science.     – George Sarton                            

Science is an effort to comprehend the world in rational terms. Rationality refers to the structured process in the brain which accepts or rejects conclusions on the basis of certain rules that seem to resonate in the normal conscious brain. The purest form of this process is reflected in mathematics, of which geometrical reasoning is a supreme example.

Euclid  (born: 27 June? 295 BCE) taught in the famed city of Alexandria in Egypt which had been founded by the young imperialist Alexander of Macedonia. Just as the Vedic rishis of ancient India made hymnal invocation to the forces of nature a canonical form of poetry, Euclid presented a formal geometrical reasoning mode that became a canonical format in geometry: consisting of definitions, postulates, and formal proofs for known geometrical results, and concluding with a triumphant QED (Quad erat demonstrandum: What was to be demonstrated has been demonstrated).

For many centuries his name came to mean in the Western world a branch of mathematics known for rigorous proofs and standard theorems. What Aristotle had done for reasoned thinking in the laws of logic, Euclid did for the theorems of geometry.

Geometry, in Euclid’s work, is not what the word literally means: measurement of the earth. Rather, it is a systematized realm in which  properties of lines, triangles, circles, and figures emerge like buildings from the foundations in the ground of postulates and  self-evident truths. The self-evident truths came to be called the axioms of Euclidean geometry.

That every geometrical form which can be drawn with  ruler and compass has some interesting properties should be clear to anyone who plays with such figures. But to recognize what these are by carefully climbing step by step on the logical ladder is more than a game in careful thinking: It is a spectacular harnessing of the powers of the human mind to walk on the tightrope of reasoning without faltering and falling. Engaging in geometrical reasoning is as fulfilling for the alert spirit, and perhaps as useless for our daily bread, as listening to glorious music. The following anecdote is appropriate. One day, when Euclid was expounding to his disciples the proof of an interesting theorem, a student is said to have raised his hand and asked how this result would come to any practical use. Whereupon Euclid threw a coin at the student, and asked him to leave the class, saying the money would fetch the wretch something useful. Those who are interested only in useful things have no business opening a book of pure mathematics or astronomy, Euclid implied.

Euclid’s classic work was The Elements. Euclid did not discover the results he published,  he established the the known results of geometry on a firm logical footing.  Other Greek writers, like Hippocrates and Leo, had published books with that title also, but they never acquired the reputation that Euclid did. The significance of Euclid’s work lies in its methodology: clear   definitions, assumptions, goals, illustrations, proofs, and conclusions. These provide a systematic framework in mathematics.

The Elements consists of thirteen Books.   Book I defines a point as “that which has no parts.” It contains the famous five postulates (aitimata: aithmata)  on which Euclidean geometry is based. The fifth of these is equivalent to saying that through a point external to a straight line only one parallel can be draw to that straight line. This came to be called an axiom or self-evident truth. Over the centuries, many tried to prove this statement, until it was discovered in the eighteen century that this is not a truth, but an assumption, as valid as the statement that the earth is flat. But assuming the earth to be flat we can draw a map of the world. This recognition led to the formulation of hyperbolic and elliptic (also called non-Euclidean) geometries in the nineteenth century.

In other words, accepting the axioms of Euclid not self-evident truths, but only assumptions one can deduce many interesting things about lines, angles, triangles, circles and such. So, in modern mathematics a postulate is a proposition we take (assume)  to be true in a give context, and from it we deduce all possible consequences by means of reasoning. What this means is that within a rational system the validity of the conclusions depends on the postulates. The conclusions need not necessarily corresponds to reality, unless the postulates do.

What is not universally recognized is that this great insight is relevant in fields beyond mathematics. Consider physics. Every theory in physics rests on one or more hypotheses. These correspond to postulates in mathematics. In mathematic, a postulate is accepted on the basis of the interesting consequences that flow from it. In physics, a hypothesis is accepted in the basis of the observable consequences that result from it.

We may extend the idea further. The notion of the postulate is also there in religion, though it is seldom recognized as such. Every religion is based on a set of fundamental assumptions which, as in Euclidean geometry, are taken to be axioms: truths, even if not self-evident.   These are the dogmas of religions, and correspond to postulates in mathematics. Just as there are different types of geometries based on different postulates, there are different religions, based on different dogmas.

But there is one important difference: Mathematicians revel in different kinds of geometry. Practitioners of religion, not realizing the postulational nature  of dogmas, regard their own system to be the only valid religious system. This is the equivalent of mathematicians claiming that only Euclidean geometry is valid. Only when this awareness  arises can there be religious harmony in the world. Who would have thought that a proper understanding of Euclidean geometry would provide some insight into the reason for religious differences?

June 24, 2011


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