The Pythagorean Controversy

C K Raju

This refers to the article on the Pythagorean Theorem by Ramkrishna Bhattacharya (Frontier, Vol 47, No 32, February 15-21, 2015). There is also an article by Prabir Purkayastha in Ganashakti http://ganashakti.com/english/comments/details/156.

The statements of Harsh Vardhan and Tharoor do need to be corrected. However, it is unacceptable to do so by appealing to just another bunch of myths. Pythagoras was the mythical head of the mystical cult of Pythagoreans. He is reputed to have been the son of Apollo, with supernatural brightness, and a golden thigh, and Apollo's own physician came to him on a golden arrow. Are people ready to believe all that just to score a political point?

It won't do to say that those myths accumulated around a historical figure. To separate myth from history one needs evidence. But where is the evidence for Pythagoras as a historical figure? What is the evidence for the myth that he proved the theorem named after him? If he did give a proof, as Purkayastha asserts, without the slightest evidence, what was that proof and from what principles did it proceed? Any compromise with the principle that history needs evidence is shortsighted.

Evidence in this context means primary evidence, which must be clearly separated from the speculations profusely added on to it by Western authorities, and propagated by Wikipedia (which tacitly accepts only Western secondary sources as "reliable"). If there is no evidence for Pythagoras or for the claim that he proved some theorem, that calls for some action. Let us agree that it should henceforth NOT be called "Pythagorean theorem", for calling it by that name perpetuates the myth. This writer agrees with Ramakrishna Bhattacharya that the "Pythagorean theorem" was known from before the 8thc to Babylonians and Egyptians. Indeed, the Elements, put together in the 5thc by a black woman, is only an exposition of Egyptian mystery geometry in the Greek language. In any case, the mythical Pythagoras certainly has no sort of claim on it.

To sustain its false history, the church uses the method of telling a thousand lies to defend one lie. Hence, the myth of Pythagoras is eventually defended by appealing to another myth: that of "Euclid". In truth there is no evidence for "Euclid" either. This author's "Euclid" challenge prize of Rs 2 lakhs for serious evidence about "Euclid" stands unclaimed for several years.

But—and here is the catch—the exposure of "Euclid" as myth did not result in any action. The myth of "Euclid" was not removed from Indian school texts written by NCERT to whose attention this was brought in 2007. The head of the NCERT math department came to a seminar organised by this writer and asked: why is primary evidence needed? He said " we go by the authority of a committee" which tacitly goes by Western authorities, like Wikipedia! Sadly, Indians colonially educated all believe exactly the same thing: they simply ignore the demand for evidence about "Euclid" and keep repeating the story thereby brazenly asserting the principle that Western myths are special and need no evidence.

That means evidence is needed only for one case, but authority suffices in another. Western myths, lacking evidence, continue to be instilled in children to cultivate biases in their mind. That creates an atmosphere where demanding evidence for Western myths is seen as an act of heresy, since it amounts to challenging Western authority, which colonial education teaches as the sole route to valid knowledge. It would be an excellent thing if those double standards go.

Now, as the quotes from the sulba sutra given by Dr Bhattacharya show, the statements in the sulba sutra-s are general statements (e.g. Baudhayana, 1.9,1.12) one of which is illustrated by particular cases. (It is an irrelevant detail that the statement is about the diagonal of a rectangle rather than the hypotenuse of a right angled triangle.) However, there is a further story aimed at the gullible who go by stories and neglect evidence. The further story is that the book Elements, which "Euclid" purportedly wrote, has a special sort of "superior" proof of the "Pythagorean" theorem, an axiomatic proof derived from axioms by reason alone, and not involving empirical observations. However, this oft-repeated claim that the Elements has some special type of "superior" non-empirical proof is (1) historically false, and (2) philosophically unsustainable.

Thus, the very first proposition of the Elements uses an empirical proof as does its 4th proposition essential to the proof of the "Pythagorean theorem". Hilariously, for 7 centuries all the best Western minds who studied the book, were so carried away by the story of their own superiority that they did not apply their mind to the very first proposition in the book which is contrary to that myth. Even more hilariously, the colonially educated blindly keep repeating that old myth without checking facts, even though those facts are known for the last one hundred years.

At the turn of the 20thc Bertrand Russell and David Hilbert in their respective tracts on the Foundations of Geometry did try to rescue the Crusading myth of "Euclid" from this predicament so embarrassing for Western scholarship (but still unknown to the colonially educated). They turned the 4th proposition of the Elements from a theorem into a postulate, the SAS postulate as today uncritically taught in school geometry. However, Hilbert's synthetic approach is unsuccessful, since it prohibits length measurements, but the "Pythagorean theorem" is a metric proposition about the equality of areas.

Anyway, the indubitable fact is that until the 20thc there was no valid axiomatic proof of the "Pythagorean theorem."

Philosophically, the claim that any kind of metaphysical proof is less fallible than empirical proofs is wrong. That claim is tied, like the myth of Euclid, to the Crusading Christian theology of reason: the dogma was that reason bound God, who could not create an illogical world, but could create the facts of his choice. Hence, reason which bound God was declared superior to facts which did not. The sulba sutra-s accept empirical proofs as does all Indian ganita. That is of contemporary value today, for it helps resist the arbitrary prohibition of empirical process from mathematics.

Do the sulba sutra-s have any other contemporary value today? In this context, it is necessary to correct an egregiously incorrect statement made by Dr Bhattacharya that "There was, however, no concept of angles and their measurement by degrees in ancient India". (Emphasis original)

First the concept. From his mention of kona, the definition of "angle" Dr Bhattacharya has in mind is the one given in current school texts. That might go as follows: "when two straight lines (or "rays") meet at a point they are said to form an angle". (That, by the way, is clearly related to a corner.) However, that school-text definition of angle is a naïve one. When one measures an angle in degrees what is it that one divides into 360 equal parts? By what process does one decide that those parts are equal? Furthermore, on this definition, it is meaningless to talk of an angle greater than 360 degrees, as one often needs to do in physics and astronomy.

In fact, straight lines are secondary to the notion of angle. The term angle primarily refers to the length of a circular arc. This can be greater than 360 degrees. An angle so defined can be and was measured with a flexible string (sulba). However, the length of a curved line cannot be measured with any instrument in the ritualistic compass box with which geometry is wrongly taught in schools today. That school geometry box has a protractor used to measure angles. If one examines the construction of the protractor, it actually divides the circumference of a semi-circle into 180 equal parts. But there is no explanation of how those parts (which are curved lines) are measured, or how they are deemed equal, without any way to measure curved lines. School students are just told to shut up and use it for that is the substance of colonial education intended to produce slaves.

Indeed, defining angles as circular arcs is not only conceptually superior, the string is an eco-friendly and low-cost substitute for the entire compass box made of steel and plastic. It can be readily used in practical contexts where the instruments of the compass box are pathetically useless. This is another aspect of the contemporary value of the sulba sutra: geometry can be taught differently, with greater practical value. In contrast, the ritualistic aspect of the geometry box is apparent: the set squares in the geometry box are never used, though most colonially educated go through school without even asking why they are there.

The unit of degrees to measure angles cornes from astronomy. This relates to the apparent motion of the sun against the background stars. "The stars turn by degrees". That is, a degree is approximately the amount by which the constellations are seen to move relative to the sun in a day (due to the motion of the earth about the sun). Hence, the measure used is 360 degrees (since this further allows the year to be approximately divided into 12 months of 30 days) during which the cycle is approximately complete. This measure is certainly found in ancient India. For instance, the RgVeda 1.164.11 speaks of a revolution of 720 (half degrees), while RgVeda 1.164.48 speaks of 360 (degrees). Of course, the Vedanga Jyotisa states that the (sidereal) year is 366 (sidereal) days. This should not to be confused with a tropical year of 365.24 tropical days, as the colonially miseducated will typically rush to do.

For measuring parts of the circumference (i.e, angle), the ancient Vedanga Jyotisa (e.g. Rk, 11) uses the term bhamsas, or amsas, which is more accurate, at about 0.1 degrees. Such sophisticated measures of angle are obviously never used in "Euclidean" geometry, and cannot even be measured with the stock protractor used in school. From ancient times, a key means of wealth in India was overseas trade which required accurate celestial navigation techniques, and hence an accurate measure of angles. Contrary to the foolish statements made by racist Western authorities like James Prinseps, Indo-Arabic navigational instruments in practical use, such as the kamal or rapalagai, used a sophisticated two-scale principle. Unlike the chauvinistically named "Vernier principle", the two-scale principle with the kamal was applied with harmonic scales, for harmonic interpolation to achieve a similar accuracy of about 0.1 degrees for angle measurements required in navigational practice. (The kamal was the instrument used by the Indian navigator who brought Vasco da Gama from Africa to India, across the "uncharted" sea.) This accuracy in Indian techniques of angle measurements is also reflected in the accurate measures of the size of the earth (by measuring the angle of dip of the rising or setting sun at equinox) found in Indian texts, from the 5thc (but not in the West, until 1672). That level of accuracy would not have been possible without an accurate technique of angle measurement. Knowledge of the radius of the earth was of great practical value for celestial navigation, hence overseas trade.

In astronomy, angle measures are also related to measures of time such as ghati, muhurta, prahara, related to the rotation of the earth about its axis, or the observed rotation of the celestial sphere. The Vedanga Jyotisa has an elaborate system of such units. (Jyotisa, by the way, means time measurement, NOT astrology. This writer has a standing public challenge from 15 years ago to show a single sentence about astrology in the entire Vedanga Jyotisa) Measuring angles through time is also the practice in modern astronomy which uses units such as the hour angle. Using time to measure angles again provides accuracy far beyond a crude protractor, for it is easy to measure angles accurate not only to minutes, but seconds and fractions of seconds. This method of angle measurement can be used to measure the radius of the earth, and the 6thc Bhaskara freely converts between longitude in terms of time and longitude in terms of distance.

To summarise, there were different ways to measure angles very accurately in Indian tradition. An angle was defined in the sophisticated way as the length of a curved line, not in the naïve way as something (what thing?) made by two straight lines meeting at a point. The reference to 360 and 720 as a way to measure revolutions is indeed found in the Rgveda, and relates to astronomy and the calendar. Texts like Vedanga Jyotisa (–1500 CE) use more accurate measures of angles in fractions of degrees. Similar accuracy in angle measurement was part of navigational and astronomical practice.

Further, instead of measuring the length of the arc (or angle) in units of the circumference (or its fractions such as degrees or amsas), for the purposes of the sulba sutra the natural measure is that of radians, or the arc (angle) measured in units of the radius. This involves the number today called pi: that is why its approximate value is calculated and given in the sulba sutras as 3.08. That is declared to be savisesa (imperfect) which involves an important philosophical principle.

This knowledge of pi improved continuously in India. The 5th c Aryabhata (Ganita 10) gives the value of pi accurate to the first sexagesimal minute (5 places after the decimal point), but declares it to be asanna (near value). Vateshwar gives sine values accurate to the second, and 14thc Madhava to the thirds, and indeed to an accuracy of 11 places after the decimal point (Nilakantha, Aryabhatiya-bhasya, commentary on Ganita 10 text and translation in this author’s Cultural Foundations of Mathematics). All this was made possible just because the sulba naturally allows the measurement of the length of a curved arc.

In sharp contrast, Europeans had great conceptual difficulty with the length of curved lines (or angles). A value of pi is attributed to Archimedes, but that is just a wildly chauvinistic claim based on 13thc Arabic manuscripts somehow attributed to his 5thc commentator Eutocius by the usual method of chauvinistic speculation. Obviously those manuscripts represent 13thc knowledge, for the Greeks in Archimedes time had no method of handling general fractions, and Europeans themselves later abandoned Roman arithmetic, acknowledging its inferiority.

In actual fact, even in the 17thc, Europeans had no clue as to how to define the length of curved lines. Descartes, a leading Western mind, declared in his Geometry that comparing a curved and straight line (as required to calculate the number pi) was beyond the human mind! It was beyond his mind surely, however, from the days of the sulba sutra, Indian children were taught how to do this by measuring a curved line with a flexible string (rajju), and then straightening it to compare with a straight line.

The most charitable interpretation of Descartes' blunder is that he was alluding to the various infinite series for pi (Yuktidipika 2.271 et seq., 2.295, 2.296, text and translation in Cultural Foundations of Mathematics, cited above) arising from the recursive method of calculating it attributed to the 14thc Madhava. The first of those infinite series corresponds to what Western chauvinists today call the "Leibniz series" for pi. The charitable interpretation of Descartes is that he thought that summing an infinite series was beyond the human mind. Newton thought he had answered Descartes' objection by saying that the infinite series could be summed metaphysically. His fluxions, however, had to be abandoned as confused, and his misunderstanding of the calculus directly led to the conceptual confusion responsible for the failure of Newtonian physics.

The relevance of the sulba stura here is this: Descartes' blunder arose due to the wrong belief that one must sum the infinite series "perfectly". Stopping after a finite number of terms, such as pi=3.14159, is adequate for all practical purposes, and is indeed the only way to make practical calculations, but it would not be "perfect", for some insignificant term of no practical value is still left out. This demand for "perfection" arose because of the Western religious hang-up with math as "eternal truth" from the days of Plato. Recall that in Plato's Meno, Socrates questions a slave boy and claims that his innate knowledge of mathematics proves the existence of the soul and its past lives. Proclus explained that Socrates used mathematics, and not geography, since mathematics has eternal truths which arouse the eternal soul (by sympathetic magic). This religious belief in the eternal truths of mathematics persisted in Crusading Christian theology on the dogmatic grounds that mathematics was the language in which God wrote the eternal laws of nature. This religious belief in eternal truths led to the demand for perfection of mathematics.

As clear from Berkeley's arguments against Newton, the accepted Western belief was that the slightest error is not to be allowed in mathematics. The idea that "perfection" could be achieved metaphysically led the West to load the calculus with a huge metaphysical structure (formal real numbers, set theory etc.) which is completely useless for its numerous practical applications. It is here that the sulba-sutra understanding of mathematics as non-eternal (anitya) and imperfect (savisesa) is useful to refute and reject that theology in present-day mathematics, and return the focus to the practical value of mathematics. This is the most important aspect of contemporary value.

Dr Bhattacharya agrees with this contributor, as do a few other Marxists, that mathematics should be taught for its practical value, and not to spread religious metaphysics.

Other related aspects of early Indian mathematics have contemporary value today. Brahmagupta's algebra used polynomials which he called unexpressed (avyakta) numbers, and that naturally led to unexpressed fractions. In formal mathematics today, these ratios of polynomials are called rational functions, which constitute a "non-Archimedean" field. Slumming an infinite series with such a "non-Archimedean" number system involves a new philosophy of discarding infinitesimals, roughly equivalent to limits by order counting, as was done in India. Corresponding philosophy may be called zeroism; it is derived from realistic sunyavada, but the new name incorporates the caveat that the important thing is practical value, not fidelity to texts, especially badly translated one's!

That "non- Archimedean" arithmetic is useful today for resolving the various problems regarding infinities arising from the Western misunderstanding of calculus which afflict contemporary physics. These problems range from the renormalisation problem of quantum field theory, the problem of shock waves and singularities in relativity, and the runaway solutions of classical electrodynamics. For more details, see the papers in the reading list posted at http://ckrai u.net/papers/Reading-list-Bengaluru.html.

In the rush to score a political point, it is essential not to get trapped in accepting the myths and dogmas erected by Crusading theologians, and instilled by colonial education to enslave minds. That education was explicitly intended as a counter-revolutionary measure. Therefore without overcoming colonial education, and discarding those myths and the related chauvinistic terminology, there is no path to revolution.

Frontier
Vol. 47, No. 34, Mar 1 - 7, 2015