Monday, October 26, 2009

History and Philosophy of Mathematics: C K Raju

Excerpts of this interview appeared in:
Ghadar Jari Hai—The Revolt Continues, Vol III, No. 3&4, July-Dec, 2009

Peepul ke Neeche

“Indian mathematics is practical whereas the European is metaphysical”

C K Raju has been arguing passionately through several lectures and books about the uniqueness of ancient Indian mathematics and how it influenced the rest of the world. He says what is taught as standard modern mathematics today, is based on theological positions taken by the Church after the Crusades. Shivanand Kanavi conversed with Raju on the results of his research in the history and philosophy of mathematics.

Shivanand: Dr Raju welcome to peepul ke neeche conversation. Having looked at some of your writings, I see that you have researched deeply into the mathematical tradition of India as well as that of Persia, Arabia and Europe. Could you give us an overview of exchanges between India and West Asia in the field of mathematics?

Raju: As I have stated in the book (Is Science Western in Origin?—C K Raju), the process of exchange with Arabs started with Barmakids (barmak from pramukh, Persian-Buddhists who were wazirs to Abbasid Khalifas--Ed), this was around 8th century CE, after the conquest of Persia by the Arabs. Besides the spread of Islam in Persia, Persian customs spread to the Arabs. There was a tradition in Persia of importing knowledge from all over the world. It was based on a philosophy which regarded knowledge itself as virtue, like the Socratic philosophy. So, to make people virtuous you gather knowledge from all corners of the world. It was begun by Khusrow Noshirvan in the 6th century. At that time Justinian closed all the schools of philosophy in the Roman empire and many philosophers took refuge in the court of Noshirvan. According to the Shahnama [of Firdausi] his wazir came to India and took chess, Panchatantra etc. back to Persia. There was also an astronomical tradition in Jundishapur (Gundeshapur) in Persia. This astronomy also traveled from India. Which is interesting, because Khusrow’s court already had the most knowledgeable people in the Roman empire and if the Almagest (Almagest is the Latin form of the Arabic name al-kitabu-l-mijisti, (The Great Book) of a mathematical and astronomical treatise proposing the complex motions of the stars and planetary paths, originally written in Greek by Ptolemy of Alexandria, Egypt, written in the 2nd century. The Almagest is the most important source of information on ancient Greek astronomy-Ed) or any other advanced astronomical text existed at that time then it would have been similarly collected and translated, but we do not hear about it. On the contrary, the Almagest itself starts off by addressing an unknown “Cyrus”. So it was probably constructed in Persia. Certainly, Greek knowledge was translated into Persian and later into Arabic. But, so far as astronomy is concerned we know that the very fact that first it went [from India] to Persia and then Baghdad shows that Greek knowledge at that point did not compare in any way with the present-day versions of Ptolemy’s Almagest. There was also a strong tradition of neo-Platonism which came through texts in Greek language [though probably it originated in Egypt]. This was called the “theology of Aristotle”, and that was the primary extent of “Greek” knowledge at that time. There was no Greek knowledge available from Byzantium at that time since all the schools of philosophy there had been closed. [We also know that Arabic knowledge travelled in the other direction, to Greek texts.] The proof is that Panchatantra is translated from Sanskrit to Pahlavi (and you find its reference in Firdausi’s Shahnama) and from Pahlavi it was translated into Arabic and then from Arabic to Greek. Among the Arabs it became the basis of a movement –Ikhwan as- Safa (the Brethren of Purity); so we know the route that knowledge took from India to Greek texts, and it also traveled directly [as in Ashoka’s time when Indian texts and medicinal plants went to Alexandria]. The process really took off with Bayt al hikma (The House of Wisdom at Baghdad) which was linked to Islamic rational theology which valued knowledge as a virtue. It was closely related to aql-i-kalaam, which meant Allah has given you aql and one must apply that aql in order to interpret the Koran.

SK: Which were the sources from which knowledge was gathered in Persia?

Raju: India was one of them. I already talked about Panchatantra, medicine. Indian mathematical texts traveled to Baghdad and they were translated by Al Khwarizmi. [Because of this movement to gather knowledge in Baghdad] the demand for books increased so much that paper technology came in from China into Baghdad. We also hear in some accounts that things came from what are called Greeks [but were from Alexandria in the African continent].

SK: Was there any exchange between Persia and Greece and Persia and India during Alexander’s (Sikander) travel through Persia up to India?

Raju: There is an account in the Zoroastrian book of Nativity that Alexander got his books from the Persian emperor and got them translated. The question is: what happened to them? Presumably, some of them [the looted books] went to Aristotle [Alexander’s teacher] and some of them went to the corpus of the library of Alexandria. Aristotle was supposed to be the first person in Greece to have a library so where did his books come from?

SK: That does not sound very different from Elgin’s marbles!

Raju: (Laughs) Yes. People have not talked about the sources of books for the library of Alexandria. It could not have been those small city states in Greece, which did not have the capacity to produce them. If you look at the trial of Socrates, there were supposed to be 600-odd jurors. If you take ten persons in the population for every citizen then there would still be only about 5-6000 people in Athens so how could they produce the books on the scale of the library of Alexandria—half a million books as is normally mentioned? Only a Persia or an Egypt could have done that.
In the case of Alexander, as with other military conquerors, knowledge flows towards them in the case of barbarian incursions.

SK: Such a large collection of books in those days must have been accumulated over a long time and must have preceded Alexander also.

Raju: Exactly! It must have taken a very, very long time. Papyrus was very expensive [so it also took a lot of resources].

SK: I said this because when I was in Deccan College, Pune, I found that they are putting together a Sanskrit dictionary and after eight volumes they are still in ‘a’since they are adding on contextual meaning as a word occurs in different canonical Sanskrit texts. They have chosen 1500 classical Sanskrit works to do so, which include natya shastra, vastu shastra, ayurveda, literary and philosophical texts and so on. If they are considering 1500 as fairly representative of canonical Sanskrit texts then to have hundreds of thousands of works it must have taken many centuries and many civilisational sources.

Raju: Exactly and that is the how the real corpus of books in the Library of Alexandria accumulated. In fact, how many Greek texts can we count? Nowhere in that neighborhood! There is no possibility that those small [Greek] states could have produced that kind of knowledge. So this entire myth making about Greeks has used this library of Alexandria. Possibly there were some texts in it that came from Greece, but nowhere in the range of half a million.

SK: There must have been Mediterranean exchanges..

Raju: The exchanges between Greece and Egypt were already taking place. Greek people like Plato, Herodotus [routinely] used to come to Egypt for higher studies. Greeks were copying Egyptian gods. Each Greek god has a counterpart in Egypt and in fact Herodotus says that explicitly.

SK: After all Egypt was a much older civilization by a couple of millennia. Did this exchange continue after the Baghdad period also?

Raju: Yes this culture of libraries spread in the entire Islamic world even in Cordoba, Spain during the Islamic period. Al Beruni when he came to India made it a point to collect knowledge of all kinds. The Baghdad book bazaar had become prominent, and this [tradition] persisted [in Islam] at least till the 12th -13th century.

SK: Arabs have been depicted as carriers and safe keepers of knowledge rather than creators of knowledge. Can you comments on that.

Raju: There is an enormous amount of evidence to the contrary. [The book mentions the case of Copernicus, where the Arabs were clearly the creators and the Europeans merely the carriers of knowledge. So] it is good to look at the question: how did this story start? (that Arabs were mere safe keepers of Greek knowledge).

SK: In fact they have been depicted as barbaric nomads killing each other, who did not have any culture till the British formed various nation states in Arabia. Thus there were Pharaohs and then there were Bedouins till the Anglo-Saxons came…

Raju: If you look at Arab literature (pre-Islamic) there is a depiction of a freewheeling society living in the desert. Post Islam, they conquered Persia and absorbed a lot of administrative structure of the Persians and then there was this culture of books and libraries. That itself shows that they had to produce books. It is a different matter that in a bazaar to get a higher price one might say not me but somebody more famous wrote this, or it was written a long time back and make it an antique etc. After all, a lot of things happen in a market. It is undeniable that Arabs were creative and made contributions so one should look at when did the story start that Arabs are only safe keepers. It started during the Crusades. They [the Christians of Europe] were fighting a religious war and Europe had a tradition of book burning. In fact, there were many fiats [by Christian emperors] right from 4th century to burn books. The library of Alexandria was burnt down. There was a tradition of burning heretical books which included secular knowledge. Within Christendom, there was not much of a culture of books and when they were fighting the Arabs they realized that they needed secular knowledge which was available in books. They captured Toledo which had a massive library [coming from] the Umayyad khilafat. It took a lot of time [for the church] to arrive at the decision to translate those books [and not burn them]. This needed a justification. That was concocted by saying that this knowledge belongs to Greece and the Greeks were theologically ‘correct’. This was regarding early Greeks mind you, since they were pre-Christian, whereas they [the church] had conflicts with later Greeks like Proclus, Theon etc. The advantage of inventing a person like Euclid was that you can attribute a philosophy to that individual which suits you.

SK: Is there any Church document or correspondence which discusses these things?

Raju: The church does not operate like that. They are not accessible. Even what the Church did in India is not accessible. If I wish to know what happened in India during the Inquisition then I don’t get access to that even if the records exist. It is not an open archive. I would rather not demand documentary evidence. In this entire [church] tradition, so many documents have been cooked up or forged. After all, even in Delhi, periodically fires go on in so many ministries and documents get burnt (laughs). Let us look at common sense and circumstantial evidence.

SK: What do you consider as Greek contributions, you have raised some questions about their arithmetical capabilities…

Raju: It is clear from their system that it was completely inadequate to do quick sums; forget about subtractions and divisions. I don’t know what their contributions were in science. I don’t have any evidence of that. May be in theatre or other things, however there is strong evidence that some ideas including Platonic ideas come from the mystery geometry tradition of Egypt.

SK: What is mystery geometry?

Raju: I have written a new book on Euclid and the mystery geometry of Egypt. If you see how Plato looks at geometry. He says it should be taught to students in his Republic, which is an ideal state. He has written about how its citizens should be trained—he particularly talks about two subjects viz music and mathematics—in order for their souls to be virtuous. The very word mathematics comes from mathesis, which means learning. What is learning? Socrates demonstrates it by calling a slave boy and asking him questions, thereby showing that the slave boy has an intrinsic knowledge of geometry. He says this is possible because the boy has a soul and the soul is recollecting the knowledge from the previous birth. In fact, the Platonic doctrine is that “all learning is recollection”. Mathematical truths are eternal, and since the soul is eternal, by sympathetic magic they [the eternal truths] arouse the soul. Thus the function of mathematics is to arouse the soul through introspection, by taking you away from the external world. This is the idea of mystery geometry. The practical applications are of no concern to us says Plato, the moral applications are more important.

SK: There are these well known names of Pythagoras and later Archimedes..

Raju: Pythagoras is a school which indulged in mystery mathematics of numbers etc. There is an exoteric part which is told to outsiders and there an esoteric part which is told to initiates. What is the evidence of Pythagoras and the proof of his theorem? [Deductive] proof is a concept post-12th century. At that time [in Pythagoras’ time] it [geometry] was only for arousing the soul. In the mystery tradition the soul knows what truth is and that [intrinsic knowledge of the soul] is the ultimate standard [of truth]. That belief about the soul came into violent conflict with [post-Nicene] Christianity, even though that notion of soul was very much part of early Christianity of Origen. From his notes the present day Bible is derived. He was declared a heretic. The doctrine of love was entirely a mystery tradition. But, after the Church and State came together in the 4th century you could not say that everyone would be saved. There had to be some advantage in becoming a Christian. It is like the state saying I am going to treat my citizens above those of other states. It brings in a boundary: this is ours and that is theirs. That is why Proclus was declared a heretic. Because he said mathematics deals with eternal truths, since the soul is eternal, therefore the cosmos should also be eternal. That goes against the [church] doctrine: for then there will be no creation and no apocalypse, so he was declared a heretic. So was the case of Hypatia and her father Theon (both prominent mathematicians from Alexandria—Ed). Clearly Christianity was uncomfortable with this interpretation of Elements and looked at it as heretical. Then Thomas Aquinas (1225-1274) reinterpreted Elements and used it as a weapon against Islam. Basically at that point in time Christendom had realized that it was not possible to spread beyond Spain by force alone. Moreover Europe was still very poor compared to the Arabs and they still coveted that money [the Arabs had]. Even though it [the Crusades] was called a religious war, it was motivated by material concerns. Like the Iraq war, which is not based on moral concerns, but on the oil wealth in the region.
Since it could not be done by warfare the church realized that it also had to adopt the method of argumentation and discourse. Quoting the [Christian] scriptures would not work with the Arabs. Thus, a third ground had to be found. That was found in the neo-Platonism that had already fascinated Islam in the form of aql-i-kalaam or falsafaa. Therefore, Aquinas realized that reason was needed to influence the Arabs. Thus, after Augustine, there was a second period of change in the Christian theological doctrine in the post-Crusade era. It was called Christian rational theology and was an adaptation of Islamic rational theology. This tried to establish universal principles of ‘proof’ [to persuade the Islamic Arabs]. That is where Elements came in.

SK: But did this not create a dichotomy within Christianity, how do you reconcile faith with reason?

Raju: It did indeed. Initially a whole lot of books ascribed to Aristotle, were banned and placed on the Index, since they were thought to be contrary to the doctrines of the Church. But then there was a whole army of people working on it who were trying to reconcile these contradictory beliefs. So it took time for “Aristotle” to be accepted into the [Christian] system. There was a process of absorption via reinterpretation. Thus Elements was reinterpreted from the tradition of mystery geometry to something which gives you a universal ‘proof’.

SK: It is like Vedanta, which says everyone is a part of the Brahman, at the same time it coexisted with the caste system…

Raju: Yes, for example there is this famous story of Shankaracharya and the chandaal, where he prostrates himself in front of the chandaal, but later it is reinterpreted. It is said that chandaal was actually a reincarnation of Shiva etc..

SK: One of the important theses put forward by you is that mathematics has cultural foundations. Can you say that there is an Indian way of doing mathematics if so what are its features?

Raju: There are some clear cut features. In India there was just one notion of proof of praman which was applied everywhere: be it philosophy, mathematics or physics. The first praman was pratyaksh. Empirical means were accepted as proof. This you find in sulbasutras, in Aryabhata, and right down to Yuktibhasha. For example the so called ‘Pythagorean theorem’ could be proved by drawing the triangle on a palm leaf, and it could be shown that the square on the ‘diagonal’ was equal to the sum of the squares on the other two sides. This could be shown by cutting, rotating etc. Whereas the European tradition would disagree and say that mathematics is purely metaphysical and by bringing in motion you are bringing in physics and it violates the basic idea of geometry as concerned with immovable space. That is one major source of tension. [Secondly], today the notion of proof is seen in a very rigid manner in a completely metaphysical way. How do you carry out deduction? on what logical basis? This is unclear in the Indian tradition. After all there are different systems of logic which are prevalent. There is the Jain system of syadvad and saptabhangi, there is Buddhist logic of chatuskoti and so on. In fact, in the debates between Naiyayikas and Buddhists over a thousand year period you find that they are not addressing each other’s issues because of differing concepts of anumaan [or deduction]. But Europeans declare their logic as universal, when it is not. There is a third aspect which I have called zeroism, which has to do with what is mathematics good for. In the neo-Platonic view it is good for the soul. The European view is that mathematics is good for providing proof. But in India, the aim of mathematics was not to provide praman but to do something vyavaharik, something practical, which is removed from soul etc. If I am doing something vyavaharik, I don’t mind making approximations. If I am computing, then the computer is going to make so many approximations. Many things are discarded or zeroed, and that is acceptable. However European mathematics demands perfection where you cannot discard the smallest entity. The belief in perfection comes from a religious view of mathematics. It then gets into theology that God made the world and he wrote the laws in the language of mathematics [which must hence be perfect]. In India it is calculations.

SK: The word for mathematics in India is ganit that is counting..

Raju: Yes it is numerical calculation. There are proofs and they can be empirical and one particular logic is not considered universal. [So proofs are not the focal concern.]

SK: When pratyaksh praman is not available you bring in inference etc. Clearly mathematics was considered something physical. Can you explain the concept of universalism that is prevalent in mathematics.

Raju: Universality is factually incorrect. The way mathematics was done in India was different from Europe. So the Indian place value system and algorithms or calculus took such a long time to be absorbed by the Europeans. Metaphysics is never universal. The moment mathematical proof becomes metaphysical it ceases to be universal. In fact it can become ‘universal’ only to the extent that it is demonstrable empirically (pratyaksh). Universality is just a European prejudice as they are ill informed about other cultures, so they declare universality from a parochial point of view.

SK: The crude way in which universality is put forward is by saying that 2+2=4, no matter where you are in Greece or Arabia, India or China…

Raju: It is not true, and I have argued it at great length in my paper presented in Hawaii. Let us say we are using a computer to add. 2+2 is a complicated case, so let us take 1+1, The answer could be 1 or even 0 depending on what kind of logic gate one is using. So, I have to specify and say I am using integers. But what are integers? If I do arithmetic with integers on computers say using a C program on a 16 bit machine it will not give 2 as the answer but something else unless I do rounding off. In order to specify what are integers I need infinite time and infinite memory. In a commercial transaction we get into an agreement saying Rs 2 plus Rs 2 would be Rs 4. But that is an agreement. It is not a universal truth. If I have two stones and if I take up two more stones then I get four stones but if I break one of them into two then I get five stone pieces. So I have to be careful about them as universal truths. At a practical level there is no problem. Even if there is no formal agreement or legal frame work, I would simply say you broke the stone. An agreement is not a universal truth or ultimate truth etc.

SK: The statement that numbers are metaphysical transcendental, entities is itself a metaphysical statement.

Raju: That is exactly the point. So long as you are in the domain of convenience it is fine. If you look at Indian texts they will have numbers with 18 digits. What will you do if you need more? you go to 20, 30 or 40 places for a particular purpose. Normally you don’t need more than 18 places. Yajurved goes only till 12 places. Aryabhat goes to 18 places. It is a matter of convenience, but you never go to an infinity of places. That is also how computer arithmetic is done. You round off after some time, and that is perfectly fine but then don’t talk about universal truth.
There is an example given in ethno-mathematics. Suppose I have borrowed two fish from you and I have returned two fish. It won’t do if I have borrowed two big fish and then returned two tiny fish. There is a sense of exchange and fairness involved, not universal truth.

SK: What is the European view on standard of proof etc.

Raju: There is the Platonic deprecation of the empirical. Then there is the clerical elevation of metaphysics over the empirical. The clergy said the metaphysical is a higher truth than the empirical truth. That is fallacious. Metaphysics is decided by a coterie.
What Hilbert did is that he analysed the Elements from this perspective: for example, the proposition 1.4 [of the Elements] or the SAS [Side-Angle-Side] theorem involves physical movement in space, like the Yuktibhasha proof of the “Pythagorean Theorem”. They said the empirical has got into mathematics, which [empirical] is perishable, not eternal, it involves motion hence physics, whereas geometry should be concerned only with properties of immovable space and so on.
So Hilbert said if this theorem is made a postulate then everything becomes metaphysical. Thus he removed the last vestiges of empirical elements in the ‘Elements’. Or at least he thought he did. But actually he could not because he had this notion of congruence which fails after proposition 1.35, the one which is used to derive the area of a triangle. There [in 1.35] congruence is not in the sense of being of the same shape but same area. Earlier propositions are about congruent triangles where you [may] just transfer attention from one shape to the other without moving them. Now [in 1.35] they are incongruent but they are equal in area. The word used in Elements is not “congruent” but “equal”. Equal again is related to equality of the soul as in say Advaita Vedanta which is also a political statement of equality of all people who might look dissimilar. The esoteric meaning is equality of dissimilar things. The way out taken by Hilbert is to define area. But how do you define area without defining length? But if you do define length then the entire Elements becomes trivial as Birkhoff showed with the metric. Thus by throwing out the empirical you start introducing peculiar and artificial things [like defining area without allowing length to be defined] Thus, Hilbert made mathematics completely metaphysical through his ‘axiomatic’ approach.
Now a lot of proofs in mathematics are based on reductio ad absurdum, which depends on two valued logic which would not be acceptable in the Indian tradition at all. So how are these proofs universal?
It is all based on and tightly tied to the [historical] perception that Aristotle the Greek did some logic etc. Of course, one does not even consider that what is called “Aristotelian logic” [might] actually have come from Naiyaikas, through the Arabs. It is a misnomer to call it “Aristotelian”. In my article on Logic for the Springer Encyclopedia of Non-Western Science, Technology, and Medicine, I have made this point that the Aristotelian syllogism is [historically] not to be found anywhere [in Greece]! There is a Stoic syllogism [in Alexandria], but then these things [Aristotelian syllogism] suddenly appear in Toledo and that is problematic.

SK: But syllogism is a very prominent part of Nyaya..

Raju: Yes that is the point, and we also know that Nyaya went to Baghdad. Anyway, the standard approach in mathematics is not universal but has been universalized. First there was the ignorance of Europeans and this ignorance has been universalized through the process of colonization. On the one side [in Americas] people are just killed off, and on the other side they are given Western education where they were given a fabricated history which made them feel inferior. The Indian elite in the 19th century swallowed this and found the solution in aping the west. This has persisted even after independence. My demand is Swaraj in science and in science education.

SK: The creative process is not deductive, otherwise rule-based machines could have done it. But post-facto deduction may be used to teach. However if again our students at the frontiers of research are not going to use the deductive approach then what is the use of even teaching this method?

Raju: Why is mathematics difficult? My answer to that is that math per se is not difficult. But if you look at the text of NCERT for 12th standard, and particularly in Hindi, you find terms like continuity, differentiability, formal real numbers, set theory etc. All this is extremely difficult to follow [in Hindi] even though I have studied all that. It is so terribly convoluted. Where are their primary axioms? They are in set theory, which enables me to axiomatically perform infinite processes, which I cannot ever hope to perform. With the axiom of choice I can have a choice function, I can claim its “existence” etc. It is only through such metaphysical and imaginary infinite processes that one can preserve the perfection of mathematics required by Western theology.
Apart from all these theological principles that have come in, you cannot teach set theory for 10th standard students, so you cannot teach the axiomatic deductive process today. I can do that only at the MSc level and very few people come to that level. The vast majority hence cannot be taught mathematics. You have to tell them a set is a collection of objects! A student has to be taught what is a ring and a field. What utter nonsense! It is very bad pedagogy.

SK: I see a great danger in this. The common perception is that Indians are good in mathematics and good with numbers. That comes from a different tradition than this abstract set theoretic one. By adopting this in our schools we are subverting ourselves!

Raju: That is right and that is the point I have made to the Knowledge Commission. Our culture has some good points and by dropping them we are subverting ourselves.

SK: A very senior executive the chairman of a large bank in Japan told me “We Japanese cannot do software because it is abstract we can do manufacturing very well. We can make things cheaper, faster, smaller etc but not deal with abstractions. Whereas Indian can do it well because they have a philosophic tradition which we lack.” I ventured to say “but you have Zen” and he just brushed it aside.

Raju: That is interesting. I hope it is true. But we are actually adopting counter cultural traditions. There is no discussion of all these things in the public space. I would like to build a quantum computer based on Buddhist logic of chatuskoti, but where is the space to discuss this?
We need to discuss what we need to teach. Somebody just sits behind closed doors and decides what should be taught and that is not correct. There is no reason. Just that we should continue to ape the west. This is how things are made ‘universal’ by a class which is educated in the western tradition and are treated as experts. If experts cannot engage in critical thinking then how do you expect the students to do it?
It is not possible to do computer arithmetic without discarding some part of a number. As soon as we start looking at what a floating point number is, we find that it is not part of a ring or a field or anything! The basic so called associative law is not obeyed. By the way, whose law? why “law”? These are all theological concepts, that the numbers must obey the law etc. All the standard algebraic structures are useless [for computer arithmetic].
In reality, there is a practical way of doing things which is embodied in the way these data types like floating point numbers are used, which is different from theoretical computer science. This encompasses a different philosophy which is closer to what I am talking about. I am talking about practical computation, where we can discard these things. But on what logic? not based on perfection or universalism! You tell me how many decimal places you need and I can procure them. That is where shoonyavad or zeroism comes in. Based on this zeroism I am conducting a course on “Calculus without Limits” in Central University of Tibetan Studies in Sarnath. I am demonstrating it to show how much simpler life can be without universalism or set theory etc.
If we say we are a secular state why should we bring in theology in mathematics, after all if I use Buddhist logic many of the theorems in mathematics will fail! We should teach secular and practical mathematics. We are doing it because the universities in the west are doing it. But those universities were erected for theological purposes. According to [Isaac] Barrow, Cambridge University was established to breed clerics!

SK: I think seeing the pragmatism embedded in western societies today I think if you build a quantum computer using Buddhist logic that can threaten the encryption involved US financial system then you would have proved your point and billions of dollars will be spent on research on alternative logic.

Raju: That is accepted. We do need to find applications, but for me the very fact that people will be able to understand much of mathematics using this new system itself would be a worth it. I don’t care if the west wants to do it or not. My son should be able to do calculations easily which he could not do earlier.

SK: I will give an example to illustrate what I was saying. Fuzzy Logic was invented by Lotfi Zadeh, a Iranian professor at University of Berkeley. There were people who called it cocaine of mathematics implying that he was high on drugs and invented this since it did not follow the normal Aristotelian binary logic. The Japanese picked it up and used it in all appliances like washing machines, TV etc. The Americans picked it up only in the 80s because they had launched an armed commando raid on Tehran in 78-79 during the hostage crisis. But the control systems of their military helicopters carrying the commandos could not stand the heat and dust of the desert. They crashed and the mission was a failure. Then they realized they needed fuzzy logic based adaptive control systems and they brought them in. In that sense they are not theological.

Raju: My concern is not to convert the west. My concern is if these theological concepts have crept into mathematics then that mathematics should NOT be taught in this country. We should teach secular mathematics. After all it is being used to condition people, inculcate inferiority in them through fake history etc.

SK: It is definitely driving people away from mathematics.

Raju: And these kids keep looking at pictures of a fake Euclid and a fake Pythagoras as white Caucasians which we see in text books, and grow up in awe of the west and say the solution to any problem is to ape the west. If we can break out of those things that itself would be an achievement.

SK: One last comment. Many have objections to the way the Indian mathematical results are written in the form of a sutra without explaining how they arrived at it or what is the justification for it. Is there any insight into how they achieved these results? Secondly, one person who wrote many results filling up many note books without giving proof is Srinivasan Ramanujan though it was in the field of analysis in the western tradition.

Raju: I am not arguing for an absence of process. To deny the value of deductive proof is one thing, and to say that there should be complete absence of process is another thing. I would assert that though there was the sutra tradition there were also Yuktibhasa, Yukti Deepika etc where they explain the process, perhaps due to Jesuit pressure! They were written after the arrival of Europeans in Cochin. A sutra has to be terse to make it easy to remember. It is a cultural matter [in the oral tradition] that here we are dealing with minds of human beings and hence the communication should be from one mind to another and not filtered through a derivation on a dead parchment where it is liable to be misunderstood. Right or wrong that seemed to have been the cultural tradition and an oral tradition. After all even Vedas are not written down. That is not a critical issue dealing with validity but a pedagogical matter.
Certainly a process has to be there and a justification [praman] has to be there. In my book [Cultural Foundations of Mathematics] I have shown [in Chapter 3] that there is complete praman for the infinite series in India, but the derivation is on different philosophical principles. I don’t say that first I should have set theory which allows me to do some infinite processes and then I should have an infinite set of numbers and then prove convergence and so on. That is the rigmarole of Western mathematics.
I want to sum the series and the stated criterion is that the sum should remain constant when I add two consecutive terms. How does it remain constant? Up to the level of accuracy and the decimal (or sexagesimal) places I need. This is similar to epsilon-delta [and the “Cauchy” criterion] but deals with a finite number of terms [and does not involve a infinite metaphysical process]. That is a perfectly good criterion.

SK: That is what physicists do when they sum any series like Raleigh-Schroedinger perturbation series. You calculate to the second order of approximation and if there is serious problem you go to the next order.

Raju: That is how all computer algorithms are done. It only ceases to be valid if you demand perfection! That is a perfectly practical attitude. It is not that process and proof are missing. It is just proof from a different philosophical position.
The first text book on philosophy that I picked up from my father said, there is no philosophical tradition in India but only poetry! For philosophy you have to read the Greeks! So now I can say that there is no mathematical tradition in Europe and it is all theology which was imported here through colonialism!
What happened with Elements is that it had come to India through Islam but it was not translated into Sanskrit till very late at the time of Sawai Jai Singh in 1723, long after the arrival of Jesuits in Jehangir’s court. There were two parallel distinct traditions. Akbar’s courtier (Abul Fazl) who wrote the Ain-e-Akbari talks about learning from the Elements. It was there in Arabic and Persian traditions but was not considered valuable by Indian mathematicians. It was considered something religious. Also, practically Pythagoras theorem comes at the end of the Elements where as Yuktibhasa starts with it, with a different way of proving it without the forty odd earlier results.
So I would say it is a religious belief which is being universalized and I find it highly objectionable. I would say, in fact, our principles are universal since they are empirical and physical. I would characterize present-day mathematics as European ethno-mathematics tainted by theology.

1 comment:

yasin yussuf said...

Prof Ck Raju is an outstanding professor. he thought history and philosophy of science at Albukhary internation university in Malaysia