... 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?

As a person Euclid is said to have been modest and kind, and an inspiring individual. He gained immortal fame, not by discovering anything himself, but by doing for geometry what Ptolemy did for astronomy: collecting, systematizing, and presenting the accumulated knowledge of his predecessors. There was a time when the study of Euclidean geometry was a requirement for becoming a high-school graduate. Times have changed. Now some young people graduate from high school without having  read a book from cover to cover, or from the first to the last screen on Kindle.

25 June 2016




“Now, therefore, the description of the Divine: Not this, not this (neti, neti); for there is no other and more appropriate description than this-not-this.”                -Yájñavalkya

Among the countless sages that adorn the pages of Hindu wisdom Yájñavalkya is, by any measure, one of the most eminent. He was prolific, unorthodox, and sometimes  rebellious to Brahmin supremacy. Yet, one of the four principal Vedas, an Upanishad, the yoga philosophy, and other important texts are attributed to him as part of Hindu canonical literature. This shows that at one time many different perspectives and pronouncements were entertained in the fertile arena of Indic philosophical discourse.

Yájñavalkya is said to have been a pupil of at least three eminent sages of the tradition. One of them – Váshakala –  is said to have come from  a non-privileged strata of society. From references to the effect that he himself ate beef and derided ritual-sacrifices, it has been suggested Yájñavalkya also belonged to that group.

There is an interesting story in the lore according to which one of his gurus once wished to conduct a ceremony to which Yájñavalkya was invited to participate. The sage not only declined, but denounced the participating Brahmin priests in uncharitable terms. The guru was so upset by this affront that he ordered Yájñavalkya to return to him all the knowledge that had been  imparted to him. Whereupon Yájñavalkya threw up the digested knowledge – so says the legend – as a dark gastric outpour. Some of the rishis present at the scene metamorphosed into partridges (tittiri), and pecked on the vomit. Then they regurgitated it as what came to be described as a Vedic compendium from partridges: Taittiriya samhita.

Yájñavalkya had two wives: Maitreyí and Kátyáyaní to whom he once wanted to bequeath all his possessions before becoming a renunciant. “Sir,” inquired the wise Maitretí, “Will I attain immortality with all the wealth in the world?” “No,” replied the man of wisdom, “There is no promise of immortality through material wealth.” “Then please initiate me into the higher knowledge which you have,” pleaded Maitreyí. This parable is to convey that loftier levels in life are not achieved through material possessions: a running theme in India’s religious-cultural history.

Yájñavalkya is credited with an important insight in theology:  The nature of the Divine cannot be described in words. Known in the Western tradition as apophatic theology or via negativa, it defines God negatively by saying He is not this, He is not that. {See the quote above from the Brihadáranyaka Upanishad (II.3.6). This leads to the notion of the Ultimate as nirgunabrahman: the Quality-less all-pervading Principle: a recognition that the finite human mind cannot conceive of infinite and unfathomable Mystery.

The Brihadáranyaka Upanishad also records the following conversation between the sage Viddagdha Shakalya (VS) and Yájñavalkya (Y). (III.9.1-26):

VS: kati deváh, Yájñavalkya? (How many gods are there, Y)? Y: As many as are mentioned in the invocatory hymns of the scriptures, which is three hundred and three, and three thousand and three. (trayas ca trí ca shatá, trayas ca trí ca sahasreti). VS: Yes, but how many Gods are really there, Y? Y: Thirty-three. VS: Yes, but how many Gods are really there, Y? Y: Six. VS: Yes, but how many Gods are really there, Y? Y: Three. VS: Yes, but how many Gods are really there, Y? Y: Two. VS: Yes, but how many Gods are really there, Y? Y: One and a half. VS: Yes, but how many Gods are really there, Y? Y: One. (eka iti.) VS: Yes, but which are those three hundred and three and three thousand and three (which you mentioned earlier)?

At this point Yájñavalkya goes on to say that those are all manifestations of the thirty-three primary gods of the Vedic framework, and then he explains who the Rudras, the Ádityas, etc. are. When Yájñavalkya comes up with large numbers for Shakalya’s question, though the answer is based on Vedic statements, the latter does not to take him seriously. This suggests that it is not always wise to take what we read in the scriptures literally. The persistent questioning by Shakalya means that one needs to probe more and more to fully understand what the core meaning of it all is.

The final answer that there is but one God is as true as the initial one that there are more than three thousand gods, because the one God is manifest in countless different forms in air and water, earth and sky, in sun and moon and stars, for God is omnipresent: The Divine is implicit in every aspect of the perceived universe. This vision of a divine  unity behind the apparent multiplicity is at the core of Hindu vision. God is too grand and magnificent to be declared as One, and just left at that. To say that the Divine conveys Truth to only one Prophet is even more restrictive of the Divine for self-expression. Manifestations of God, whether as minute atoms or  mammoth stars, as mindless animals or thinking humans, as poets or prophets, are infinite.

Yájñavalkya was the first thinker in history to articulate a view of consciousness. In a dialogue on how we come to know about things in the world, he talks of the sun and moon fire and speech, and final to an inner light (the self) as the source of all awareness. In no other writing before this one (7th century BCE)  had there been even a mention of the ultimate source of knowledge.

Here was an extraordinary thinker about whom we know very little: where he lived and when or how he transmitted his insights. As per the lore, he was in the court of King Janaka of Ramayana. He is also reported to have been at the Rájasúya sacrifice performed in the Mahabharata. The two epics transpired in different eons.  The mixture of fanciful legends and anecdotal hearsay euphemistically called sacred history, give us only some legends: interesting but clearly imaginary..

22 June 2016