Marx and mathematics4:
The epistemic test
C. K. Raju
So, what difference does a false history of calculus make to the issue of Marx’s understanding of calculus? Or for that matter to our understanding of calculus? As we saw in part1, the key difficulty that Marx had in understanding the calculus was the fact that no one in Europe properly understood the calculus in the 19th c. Why not?
My epistemic test is the first step to understand this widespread lack of understanding. As a teacher, one often comes across students who cheat in an exam, and turn in answer sheets which are nearly identical. The students, of course, invariably deny having cheated. So what is the remedy? Test their understanding again: ask searching questions to the students to check whether they can explain the answers they wrote down. Those who wrote down answers without full understanding have clearly copied from others. In a word, those who copy from others fail to understand what they copy, and hence failure to understand is proof of copying.
This epistemic test applies even in cases of mass copying. Once when I headed a gargantuan department, with some 38000 students, I had to examine answer sheets in a case of mass copying: though the answers were nearly identical (they had been publicly dictated) each answer sheet contained individual mistakes which were obvious to the experienced eye. Those mistakes revealed lack of knowledge of important issues. I have applied this epistemic test also to people appearing for a jobinterview for the position of a full professor, who are unable to explain sentences in their own Ph.D thesis!
The epistemic test exposes not only cheats in exams, it also exposes cheats in history who lack understanding of what they claim to have discovered. While cheats in exams are generally bad students, in the case of history the people involved may have become famous by falsely claiming other people’s intellectual achievements as their own. But their lack of understanding nevertheless exposes them to the knowledgeable.
The most famous of these is Einstein, who copied the special theory of relativity from Poincare, without understanding (till the end of his life) what Poincare understood, that relativity forces the use of functional differential equations. The falsehood of Einstein’s claim to have discovered relativity is exposed by his lack of understanding (until the end of his life) that functional differential equations are fundamentally different from the ordinary differential equations of Newtonian physics.1 But Einstein is hardly the sole example—this is a Western habit.
While correcting this “Einstein’s mistake”, in my above book, I pointed out that the use of the correct type of functional differential equations may explain also the mysterious features of quantum mechanics. Amusingly, a successor of Newton, as President of the Royal Society, and a famous mathematician, Michael Atiyah, copied my thesis (during a special lecture on the centenary of Einstein’s relativity paper!) and claimed it as his own, without acknowledging my previously published work. In the true tradition of brazenly false Western history, he persisted in claiming credit without acknowledging my work, even after he was personally informed of my published books and articles, for he believed he could get away easily given the slavish credulousness of the colonised, and the support for him in the West. That is, he made a second attempt to claim my thesis as his own, again without acknowledging my past work, when it was beyond all doubt that he knew of my past work. However, the thief was caught by the epistemic test.
He went by the observation that people go along with any nomenclature without investigating its historical validity. Hence, he tried to attach his name to my idea, and got some sycophants to coin the term “Atiyah’s hypothesis”. This was also socially savvy compared to my term “Einstein’s mistake” (which provokes people who find the attack on their false idols offensive). But this savvy term “Atiyah’s hypothesis” exposed his lack of understanding: Atiyah did not understand that I made no hypothesis, but only argued that the math related to existing physics must be done correctly.
So, to reiterate, the epistemic test applies not only to bad students, but also to the false stories told by those who are today glorified as the leading mathematicians and physicists in the world. Similarly, Copernicus made mistakes while copying his heliocentric thesis from the work of Ibn Shatir, a Greek translation of which was available in the Vatican library.
Though Newton and Leibniz are hero worshipped by the colonised, they claimed credit for this stolen knowledge of calculus only on the infamous and genocidal doctrine of Christian discovery, according to which only Christians can be “discoverers”. But UNDERSTANDING is a basic condition for genuine discovery, and the fact is that neither Newton nor Leibniz understood how to sum infinite series, or what exactly the derivative was. In fact, due to fundamental differences between the Indian and the Western understanding of mathematics, this state of affairs persisted.
A similar thing happened with zero: the West took centuries to understand it because of the difference between their primitive system of the abacus (tied to Roman numerals), and the sophisticated Indian arithmetic of algorithms, which it called “Arabic numerals”. Anyway, there was no coherent Western account of calculus in Marx’s lifetime, not until the 20th c., when formal set theory was established enabling a semicoherent (but metaphysical) account of formal “real” numbers. In particular, it would be unfair to blame Marx alone for not understanding calculus: no one in Europe properly understood it in the 19th c.
That is, the epistemic test resolves the mystery of the specifically European magic of “discovery without understanding.” The claim of discovery is fraud, which is exposed by the truth of lack of understanding.
Practical value?
Now, of course, it could be argued that the true test is practical value. But such an argument would go against the grain of Western chauvinistic history: there is not the slightest doubt that black Egyptians obtained practical value from the “Pythagorean theorem” which is credited to Pythagoras on the (false) claim that Pythagoras had a better understanding. (We will not go into the truth of the claim right now.) And in the case of the calculus, the fact is that Indians had obtained practical value from the calculus (e.g. for navigation), long before the West. So, emphasis on practical value works against the West. As we will see, Newton’s critic Berkeley also emphasized understanding, claiming there can be no science without proper understanding.
Further, contrary to what many might believe, practical value has nothing whatsoever to do with formal real numbers and limits which the West and current university education associates with the calculus. Thus, even today, practical applications of calculus are all done using finite differences and numerical methods (especially calculations of rocket trajectories by NASA, a case which is often cited by the ignorant). An easy way to understand this is that these calculations are today done using computers which CANNOT use unreal real numbers or the metaphysics of limits. In fact, computers use floating point numbers, which are even algebraically different from formal real number, for they do not even “obey” the associative “law” for addition.2 Hence, what is done on computers, involves inexact numerical methods. But this was exactly the method of inexact but precise numerical calculation that was first proposed by Aryabhata,3 and used for over a thousand years in India, before it was stolen by Euler, 1200 years later, and came to be called the “Euler” method of solving differential equations, since Western historians, as usual, falsely give credit to the laterday Euler.
Unfortunately, Marx just believed all this fraud Western history, without checking it, and did not ask how could the subject have been discovered by those who lacked a clear understanding of it. In contrast, Indians had a very clear understanding. To obtain his precise trigonometric values, Aryabhata himself used only finite differences (which method Marx praises in contrast to what he calls the mystical calculus of Newton and Leibniz, but wrongly attributes to D’Alembert). Further, Aryabhata did not need derivatives: he just used the elementary rule of three (taught in primary school) or linear interpolation, which he turned around into linear extrapolation or the “Euler” method, 1200 years before Euler.
In any case, it is beyond doubt that the precise Indian trigonometric values which the Europeans coveted for their practical value in navigation, were obtained centuries earlier in India, by an extension of Aryabhata’s 5th c. method by the Aryabhata school in Kerala.
Newton’s fluxions
So, we need to go back to the question of “understanding”, and reexamine Newton’s claim to have developed a “rigorous” understanding of calculus through the use of “continuous” fluxions, as distinct from Cavalieri’s discrete indivisibles. By way of information, (1) fluxions are today abandoned as utterly confused, and, to reiterate, (2) practical value from calculus is mostly obtained today using computers, which cannot handle the metaphysics of the socalled “continuum” or formal real numbers, and make do with discrete floating point numbers.
Specifically, the European lack of understanding of calculus in the 18th c., even after Newton’s death, was well understood by Europeans, and is illustrated by Berkeley’s arguments against Newton. Admittedly, Berkeley’s concerns were political: he was afraid that Newton’s 7volume lifetime secret research on church history, pinning down the numerous church forgeries and falsehoods in the Bible, would leak out and bring enormous disrepute to the church.4 Newton, though a fanatic Christian, was totally antichurch, and in his secret notes he called church priests “spiritual fornicators”, “the most evil sorts of men ever to have inhabited the earth” etc. (In particular, Newton’s conflict with the church was NOT a case of “science vs church” as so many colonised minds will wrongly jump to guess based on the stories they have internalised.) It was a case of simple religious hostility: the hostility of a devout Christian against the highly politicised church.
Newton’s religious hostility to the church (for having corrupted the Bible), as articulated in his secret writings across fifty years, resulted in its suppression. An assessor from the Royal Society described that lifework of Newton as “foul papers related to church matter, unfit to be published”. (Those papers are still a safe secret, even though Newton’s rough drafts leaked out in the late 1960’s forcing a rewrite of Newton’s biography.5 ) However, the Cambridge biographer of Newton, D. T. Whiteside, in a spat with me in 2000, in the Historia Matematica list, was still trying to make people believe the fraud that Newton’s works were all published, though by this time, the scandal of the suppression of Newton’s work was widely known following the Imperial College’s 1998 public announcement of trying to recover those suppressed papers of Newton. The low standard of Cambridge history is clear from the fact that Cambridge historians could not write an honest history of one of their own in three centuries. Anyway, as a bishop and representative of the church, Berkeley hence titled his article against Newton “a discourse addressed to an infidel mathematician”.6
Irrespective of Berkeley’s motivation to pull down Newton, and irrespective of his polemics, his arguments are valid. He is careful to point out that he is not challenging Newton’s scientific conclusions, but only his understanding. He explains in detail how one can arrive at a right result in a wrong way, through a double mistake each of which cancels the other. His point was that such a process, lacking understanding, cannot be called science: “For Science it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means.”
At the heart of the matter is Newton’s notion of “Fluxion” today abandoned as incredibly confused, but then the centrepiece of Newton’s wrong claim to rigor in calculus. As Berkeley put it:
And. . . there be other Fluxions, which Fluxions of Fluxions are called second Fluxions. And the Fluxions of these second Fluxions are called third Fluxions: and so on, fourth, fifth, sixth, &c. ad infinitum. Now as our Sense is strained and puzzled with the perception of Objects extremely minute, even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein. . . And it seems. . . to. . . exceed, if I mistake not, all Humane Understanding. The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered; the Objects, at first fleeting and minute, soon vanishing out of sight. Certainly in any Sense a second or third Fluxion seems an obscure Mystery.... the nascent Augment of a nascent Augment, i.e. of a thing which hath no Magnitude: Take it in which light you please, the clear Conception of it will...be found impossible.
Unfortunately, in the West, truth is often totally tied to immediate political requirements. Therefore, once the potential political crisis for the church ended, with the successful suppression of Newton‘s 7volume history of the church, Berkeley’s criticism was ignored. But the fact remains that Berkeley was right: Newton did not understand calculus, and his supporters could not defend him against Berkeley’s critique.
For example, in a heated response to Berkeley, James Jurin7 haughtily “clarified”:
A nascent increment is an increment just beginning to exist from nothing, or just beginning to be generated, but not yet arrived at any assignable magnitude how small soever.
This is just a meaningless barrage of words, demonstrating complete incomprehension. In particular, we should not commit the fallacy of retrospective reinterpretation in a favourable light of terms such as “limits” etc. Naturally, Marx, too, was lost in trying to understand what exactly a derivative/fluxion was.
As Berkeley further points out the infinitesimal is equated to zero at the end of a calculation, so why not do so in the beginning itself? Marx does something of the kind equating dy/dx to 0/0, or at least so it reads in the version to which I have referred.8 He naturally also speaks of the mystical differential calculus recognizing the difficulties in understanding it (from the European perspective). In India, what was used was always the finite differences (e.g. khandajya) never any metaphysical limiting notion of derivative, which is not needed for ANY practical application.
Perhaps the ultimate argument for Newton’s lack of understanding of the calculus is the failure of Newtonian physics on theoretical grounds,9 that Newton made time metaphysical, and hence Newtonian physics had to be replaced by relativity. Newton made time metaphysical because of his superstitious belief, like Galileo’s, that God had written the (eternal) laws of nature in the language of (eternal truth) mathematics, so it was important for mathematics to be perfect.
To conclude, the epistemic test resolves the mystery of “discovery without understanding”. Newton’s claim of better understanding of calculus, or rigour with his fluxions, was actually completely confused, but this claim had no relation to the practical value of the calculus which is still obtained using only finite differences.
So, does that settle the issue of “understanding” of calculus? Not quite. There are several things that have been left out. Thus, there is (1) the question of whether the current (university) understanding of calculus based on limits and real numbers is satisfactory, even within formalism. (It is not.) There is a more fundamental question (2) whether the formalist philosophy of mathematics itself is at all a good philosophy of math. (It is not.) Thirdly, there is the question (3) what alternative there is, whether the philosophy with which calculus developed in India is a suitable alternative for teaching and advanced technological applications of calculus. (It is.) But a discussion of all this would take us too far away from the immediate issue.
Let us instead move on to the last issue we will consider, which is this: granting that Marx did not understand the calculus in his time (or even granting that the West still doesn’t understand it), what difference does it make to any aspect of Marxism?
(To be continued)
Notes & References
1 C. K. Raju, Time: Towards a Consistent Theory (Springer, 1994). For a simplified account see ; C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003). An easily available account online is in C. K. Raju, “Functional Differential Equations.1: A New Paradigm in Physics”.” Physics Education (India) 29, no. 3 (July 2013): Article 1. http://physedu.in/uploads/publication/11/200/29.3.1FDEsinphysicspart1.pdf. “Functional Differential Equations 2: The Classical Hydrogen Atom”.” Physics Education (India) 29, no. 3 (July 2013): Article 2. http://physedu.in/uploads/publication/11/201/29.3.2FDEsinphysicspart2.pdf.
2 For an example, and a computer program to demonstrate the failure of the associative law for floats, see C. K. Raju, “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā,” Philosophy East and West 51, no. 3 (2001): 325–362.
3 C. K. Raju, Cultural Foundations of Mathematics (Pearson Longman, 2007) Chapter 3, “Infinite series and pi”,.
4 Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs chp. 3 “Newton’s secret.”
5 Richard S. Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge University Press, 1983).
6 George Berkeley, The Analyst or a Discourse Addressed to an Infidel Mathematician (London: J. Tonson, 1734), http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.html.
7 James Jurin, Geometry no Friend to Infidelity, London, 1734, and The Minute Mathematician, London, 1735.
8 (trans.) Pradip Bakshi, Karl Marx: Mathematical Manuscripts, (Kolkata: Vishvakos Parisad, 1994) First edition.
9 Time: Towards a Consistent Theory, cited in reference 1 above.
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Sep 8, 2020
Prof. C. K. Raju ckr@ckraju.net
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