The issue was Newton's unpublished papers. Ryan Castle wanted to know whether some of Newton's papers are still unpublished. In response, I said, "a 'cartload' of Newton's papers (his lifework, in fact) is unpublished"; some 4 hours _later_ Whiteside stated "NONE-repeat NONE-of Newton mathematical manuscripts remain unpublished". Given the gravity of the issue for historiography, and its implications for the reliability of the present-day history of science, the incidental juxtaposition of our two responses created a striking contrast between "'cartload'" and Whiteside's suggestion of "NONE...". The next day, Whiteside described my response in the following terms: "I find this incredible...when I am roused I go straight for the jugular...Get real C. K. Raju". Though the response made evident Whiteside's fury, there was noticeable in it a _complete_ absence of the slightest reference to any specific facts or arguments. Accordingly, I invited his specific corrections to my fact s and arguments, something to which I am always open.
Despite the prolixity of his latest response, Whiteside has again failed to _specify_ exactly what it was in my brief response (to Ryan Castle's question) that could conceivably have been labeled as "incredible" or "unreal". The unavoidable inference is that, despite the strength of the adjectives, there was nothing in my response (of 20 December) with which Whiteside could specifically disagree--nothing real to which he is able to attach those adjectives! With what purpose, then, were they suggested? and whose credibility does that reflect upon, Tom Whiteside?
Was it the existence of Newton's unpublished papers that Whiteside found incredible? So his response suggested; however, he (not I) was soon corrected, and he now explicitly admits the existence of Newton's unpublished papers. More precisely (since he now realizes that my reference to the Newton project at Imperial College was enough to obtain information on Newton's unpublished papers at the click of a mouse button), he now has little option but to explicitly concede also the huge bulk of Newton's unpublished papers.
Having himself been forced to get a little more real, and in striking contrast to the amazing posture of incredulousness that he had earlier struck, all that Whiteside now _seems_ to contest is whether Newton's unpublished papers constitute a "cartload" or an "armload". I imagined he was knowledgeable enough to recognize which authority the "cartload" (originally in quotes) alluded to. Since this is the only tangible point on which he _seems_ to correct me, am I to believe that here too he blundered in not spotting the allusion (by sloppily omitting the quotes) and thus inadvertently misdirected his criticism? In that case, he might want to read my book to learn more (if he at all regards that as possible). However, it seems to me rather more likely that while plucking the "cartload" out of quotes, Whiteside has also plucked it out of context. (In Indian terminology, this mode of argumentation is called vitanda; I have been pointing out for many years that this way of makin! g stat ements that _suggest_ one thing, but can be retrospectively disambiguated to mean another, is a favourite method used by astrologers and other purveyors of untruth.) Quite possibly, the particular lot of unpublished Newton papers to which Whiteside refers constituted an armload, but the "cartload" referred to the entire undivided bulk of Newton's suppressed papers, including drafts etc., some of which later became part of the Yahuda collection, now in Jerusalem. Whichever the way we interpret Whiteside's response to "cartload", the conclusions are not happy.
I should add that I hold no brief for any authority, and I am perfectly willing to let the issue of "cartload" vs "armload" be decided empirically. But let Whiteside first state something clearly _refutable_ about the exact undivided bulk of Newton's unpublished papers--for, now that he admits knowledge of their existence, the inference naturally is that Whiteside has also gone through most of Newton's unpublished papers to be able to say confidently that NONE of them pertains to mathematics.
The point about refutability is important since excuses destroy refutability, and Whiteside's response lists some generic excuses that authorities (like Brewster?) might have had for knowingly not writing the truth about Newton. There seems to be a misunderstanding here. I was not contesting the availability of such "good" excuses to authoritative Western historians for persistently not telling the truth about Newton for some 250 years. Since knowledge of Newton's papers was hidden for so long, doubtless there are many many excellent excuses, some of which may well be genuine. I was merely asserting that (a) the truth about Newton had been suppressed for centuries by Western historians of science, and (b) that this suppression serves as an indication of the reliability of the history they wrote. I abide by these assertions, and I regard as misguided Tom Whiteside's persistent attempt to sidestep these key issues of history and historiography by starting a new thread of pers! onal a buse against me!
Indeed, I had invited Tom Whiteside to shift from polemic to facts and arguments. Instead, in his response, he has shifted in the other direction of being prolixly abusive! Perhaps he subscribes to the belief that the stronger the abuse, the higher his authority, and hence the better the argument ! This sort of intemperate response is not suggestive of a sincere desire to arrive at the truth. Further, if one were to accept Whiteside's claim that this abuse is not specifically racist, then it seems to follow that Whiteside commonly engages in such abusive debate, with his colleagues for example, so that this represents the tradition of debate at Cambridge. I rather doubt that, though I am aware that a key concern there was, in Isaac Barrow's words, with "the breeding of clerics", hence propaganda. In my university, however, we teach undergraduates, to the contrary, that abuse is a sure indication of a weak argument---the stronger the abuse the weaker and more ineffectual the ! argume nt. Indeed, the tradition of debate in India was concerned with eliciting the truth, so that the ancient Nyaya Sutra lists abusiveness as one of the definite ways of losing an argument. Further, as the Buddha pointed out, when he was confronted by an abusive person, if a guest comes to your house, and you offer him food which he does not accept, the food remains yours. Likewise, if I don't accept the abuses you offer, they remain yours, Tom Whiteside, for you to chew and reflect upon the real causes of their origin, in your attempt to impose your authoritative version, despite your persistent inability to respond with facts or arguments to the sad story of Newton's secret.
Whiteside's abusive response might, however, serve to clarify an important sociological issue, for, unlike the celebrated case of Galileo, which Protestants in the 17th c. CE made much of, Newton's case involved no clear central religious authority. So how was Newton's work successfully suppressed for so long, with so few people even getting to know about it? Whiteside's response makes manifest the reasons _why_ Newton chose to conceal the truth of his views during his lifetime--presumably to defend his jugular from those who would have thirsted for his blood--and why other scholars remained afraid of articulating that truth and suppressed and misrepresented it for another 250 years after Newton's death. (The oft-repeated statement, reiterated by Whiteside, that Newton's theological concerns predominantly related to his old age was, I thought, long ago conclusively shown to be false by Richard Westfall, in his revised biography of Newton, "Never at Rest", published some 20 y! ears a go, and I thought that few scholars today would disagree that Newton's writings on theology far exceed in bulk his writings on physics and mathematics. Incidentally, Westfall too complains in that book, of being denied access to the final version of Newton's "History of the Church".) Understandably, the fear of such abusiveness or deliberate misrepresentation or personal harm may have forced into silence Newton who was surely aware of the even longer history of how such tactics had often been "successfully" used by theologians to silence critics long before him--indeed, this was the very thing that Newton hoped to expose in his suppressed writings! Despite the evidence for the long-term "success" of these tactics, I refuse to be cowed down by them, and such tactics will not induce me to accept as legitimate the Western monopoly on historical "truth". However, even if I were thus forced into silence, what truth would that have established about the authoritative history of sc! ience except that these grand "truths" have no other viable basis?
The origin of the calculus ==========================
I mentioned that, from such a tradition of history-writing, numerous subtler distortions were to be expected, just like Whiteside, while conceding Madhava's priority for the development of infinite series, distorts the dates of both Madhava and the Yuktibhasa, by about a century in each case. (Madhava was 14th-15th c. CE, not 13th, while the Tantrasangraha [1501 CE] and Yuktibhasa [ca. 1530 CE] are both 16th c. CE texts, not 17th.) In fact, in the 16th c. CE Jesuits were busy translating and transmitting very many Indian texts to Europe; during the 16th c. CE, their activities were especially concentrated in the vicinity of their Cochin college, where they were teaching Malayalam to the local children (especially Syrian Christians) whose mother tongue it was, and where copies of the Yuktibhasa and several other related texts were and still are in common use, for calendar-making for example. [The Jesuits, of course, needed to understand how the local calendar was made, especi! ally s ince their own calendar was then so miserably off the mark, partly because the clumsy Roman numerals had made it difficult to handle fractions. Moreover, European navigational theorists like Nunes, Mercator, Stevin, and Clavius were then well aware of the acute need not only for a good calendar, but also for precise trigonometric values, at a level of precision then found only in these Indian texts. This knowledge was needed to improve European navigational techniques, as European governments desperately sought to develop reliable trade routes to India, for direct trade with India was then the big European dream of getting rich. At the start of this period, Vasco da Gama, lacking knowledge of celestial navigation, could not navigate the Indian ocean, and needed an Indian pilot to guide him across the sea from Melinde in Africa, to Calicut in India.] After the trigonometric values in the 16th and early 17th c. CE, exactly the infinite series in these Indian texts started appe! aring in the works, from 1630 onwards, of Cavalieri, Fermat, Pascal, Gregory etc. who had access in various ways to the Jesuit archives at the Collegio Romano. Since Whiteside has a copy of the printed commentary on the Yuktibhasa, he could hardly have failed to notice this similarity with the European works with which he seeks to make the Yuktibhasa contemporaneous!
I have no doubt that in the course of "the fabrication of ancient Greece" (in Martin Bernal's words), some Western historians acquired ample familiarity with this technique of juggling the dates of key texts. Having anticipated this, the evidence for the transmission of the calculus from India to Europe is far more robust than the sort of evidence on which "Greek" history is built--it cannot be upset by quibbling about the exact date of a single well-known manuscript like the Yuktibhasa.
While the case for the origin of the calculus in India, and its transmission to Europe is otherwise clear, there remains the important question of epistemology ("Was it really the calculus that Indians discovered?"). For, while European mathematicians accepted the practical value of the Indian infinite series as a technique of calculation, many of them did not, even then, accept the accompanying methods of proof. Hence, like the algorismus which took some five centuries to be assimilated in Europe, the calculus took some three centuries to be assimilated within the European frame of mathematics. I have examined this question in depth, in relation to formalist mathematical epistemology from Plato to Hilbert, in an article "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa", Philosophy East and West, 51(3), 2001, 325--61. (That paper, incidentally, proposes a new understanding of mathematics, for it argues that formal deductiv! e proo f does not incorporate certainty, since the underlying logic is arbitrary, and the theorems that can be derived from a particular set of axioms would change if one were to use Buddhist logic, or, say, Jain logic; any responses to this are welcome through the more formal medium of print.)
Indeed, I should point out that my interest in all this is _not_ to establish priority, as Western historians have unceasingly sought to do, but to _understand_ the historical development of mathematics and its epistemology. The development of the infinite series and more precise computations of the circumference of the circle, by Aryabhata's school, over several hundred years, is readily understood as a natural consequence of Aryabhata's work, which first introduced the trigonometric functions and methods of calculating their approximate numerical values. The transmission of the calculus to Europe is also readily understood as a natural consequence of the European need to learn about navigation, the calendar, and the circumference of the earth. The centuries of difficulty in accepting the calculus in Europe is more naturally understood in analogy with the centuries of difficulty in accepting the algorismus, due, in both cases, to the difficulty in assimilating an imported e! pistem ology. Though such an understanding of the past varies strikingly from the usual "heroic" picture that has been propagated by Western historians, it is far more real, hence more futuristically oriented, for it also helps us to understand e.g. how to tackle the epistemological challenge posed today in interpreting the validity of the results of large-scale numerical computation, and hence to decide, e.g., how mathematics education must today be conducted.
I would not like to go further here into the difficult question of epistemology, and the interaction between history and philosophy of mathematics, except to link it to Whiteside's use of the phrase "Hindu matmatics" [sic]. Am I to understand that Whiteside now implicitly accepts also the possible influence of Newton's theology on his mathematics, and is alluding, albeit indirectly, to some subtle new changes brought about by Newton in the prevailing atmosphere of, shall we say, "Christian mathematics"? Probably not. I presume instead that, despite his protestations to the contrary, Whiteside is really referring to the Eurocentric belief that there is only one "mainstream" mathematics, and everything else needs to be qualified as "Hindu mathematics", "Islamic mathematics" etc.
Now it is true that I have commented on _formalist_ mathematical epistemology from the perspective of Buddhist, Jain, Nyaya, and Lokayata notions of proof (pramana), in my earlier cited paper and book. I have also commented elsewhere, from the perspective of Nagarjuna's sunyavada, on the re-interpretation of sunya as zero in formal arithmetic, and the difficulties that this created in the European understanding of both algorismus and calculus, difficulties that persist to this day in e.g. the current way of handling division by zero in the Java computing language. Nevertheless, having also scanned the OED for the meaning of "Hindu", I still don't quite know what this term "Hindu" means, especially in Whiteside's "ruggedly individualistic" non-Eurocentric sense, and especially when it is linked with mathematics! Given the fundamental differences between the four schools listed above, it is very hard for me to dump them all, like Whiteside, into a single category of "Hindu"; on the other hand, if we exclude some, which counts as "Hindu" and which not, and why? and exactly how does that relate to mathematics?
A key element of the Project of History of Indian Science, Philosophy, and Culture, as I stated earlier, is to get rid of this sort of conceptual clutter, authoritatively sought to be imposed by colonialists (and their victims/collaborators), and to rewrite history from a fresh, pluralistic perspective. In my case, it is part of this fresh perspective to redefine the nature of present-day university mathematics by shifting away from formal and spiritual mathematics-as-proof to practical and empirical mathematics-as-calculation. Since my objective is truth and understanding, I am ever willing to correct myself, and I remain open to all legitimate criticism, but I do not recognize dramatic poses, assertions of authority, abuse, cavil, misleading circumlocutions, etc. as any part of such legitimate criticism.
There are numerous other points in Whiteside's prolix response, to which it would be inappropriate to provide detailed corrections here. [E.g., I do not share the historical view needed to speak of the "re-birth" of European mathematics in the 16th and 17th c., which view Whiteside freely attributes to me, though I would accept that direct trade with India in spices also created a direct route for Indian mathematics, bypassing the earlier Arab route.] For the record, I deny as similarly inaccurate all the interpolations and distortions he has introduced into what I have said.
There is, however, one issue which remains puzzling, even from a purely Eurocentric perspective. In what sense did Newton invent the calculus? Clearly, the calculus as a method of _calculation_ preceded Newton, even in Europe. Clearly, also, the calculus/analysis as something epistemologically secure, within the formalist frame of _mathematics as proof_, postdates Dedekind and the formalist approach to real numbers. While Newton did apply the calculus to physics, that would no more make him the inventor of the calculus than the application of the computer to a difficult problem of genetics, and possible adaptations to its design, would today make someone the inventor of the computer. Doubtless Newton's authority conferred a certain social respectability on the calculus. The credit that Newton gets for the calculus depends also upon his quarrel with Leibniz, and the rather dubious methods of "debate" he used in the process. But none of this convincingly establishes the credi! t for calculus given to Newton, even within the Eurocentric (as distinct from Anglocentric) frame. So what basis is there to give credit to Newton for originating the calculus, while denying it, for example, to Cavalieri, Fermat, Pascal, and Leibniz?
C. K. Raju
Professor & Head Centre for Computer Science MCRP University, Bhopal
Editorial Fellow Project of History of Indian Science, Philosophy and Culture Centre for Studies in Civilizations New Delhi