Did you find math difficult in school? Does your child? If so, what is the solution: change the teacher or change the child? Blaming the teacher or the child for math difficulties is a common but unsound explanation. Thus, problems with teachers or students should equally affect all subjects, not only math.The right solution is to change math. That seems impossible. People naively believe that math is universal. In fact, the math taught today, from middle school onward, is called formal math; it began only in the 20th c. with David Hilbert and Bertrand Russell. It differs from the normal math which people earlier did for thousands of years, across the world, and still do in kindergarten.Formal math adds enormously to the difficulty of math but nothing to its practical value. The practical value of math comes from efficient techniques of calculation, used in normal math, not prolix formal proofs. For example, the proof of 1+1=2 took Whitehead and Russell 368 pages of dense symbolism in their Principia. That proof is a liability in a grocer’s shop. In contrast, normal math is easy. One apple and one apple make two apples as most people learn in kindergarten. So should we switch back to normal math at all levels?
Now our school texts justify the teaching of formal math as follows. The class 9 math text of NCERT1 (or of various states). tells the story of an early Greek called Euclid who was the first to do math “systematically” using deductive reasoning. The text further asserts that this was something that all others in Egypt, India, Iraq, China, and South America failed to do. It shows children an image of Euclid as a white man. On this story, students are told, we must do math by imitating “Euclid” who is glorified as the father of “real” math (meaning formal math).
The story condemns normal math as inferior. But no real argument is advanced to support the purported superiority of formal math, just a story. Do children check it? No; but the story is false. To expose its falsehood, I have offered a prize of Rs 2 lakhs for any serious (primary) evidence about Euclid. This prize stands unclaimed for several years. Why? Experts know2 there is nil evidence for “Euclid” and much counter-evidence.
Our own “experts”—the one’s who wrote the NCERT text—are unable to produce the evidence when challenged. They should either accept their mistakes, or defend their claims publicly, but do neither. Since we are totally dependent on such “experts” we just carry on with the wrong school texts! The vested interests involved are, however, deeper than just the vanity of “experts”, or their desire to preserve their jobs. Hence, attempts to publicly challenge the story of “Euclid” or to challenge the related philosophy of formal math are often censored.
As just one example of censorship, I wrote an article, “To decolonise math, stand up to its false history and bad philosophy”. This was published in the Conversation (global edition) in Oct 2016. The article created a stir. It went viral and recorded some 17K hits (60% in US and Africa) before it was abruptly removed by the South Africa editor. If there was something wrong in the article, the Conversation should have carried a public correction. No one was able to point to any actual error. So, the removal was privately justified on the lame editorial ground that I had “sited” (sic) my own work, such as my book.3 Even in India, the article was first reproduced and then taken down by both The Wire and Scroll, though to the credit of The Wire it put the article back with an apology. Currently, that censored article is available on my blog,4 on The Wire,5 and on Science2.0.6 It was also recently reproduced in full as part of another peer-reviewed journal article,7 so, again, there was nothing obviously wrong with it.
So, why was it censored? Why are false myths and censorship so essential to the teaching of math?
The answer involves three unpleasant facts. First, this way of teaching math came to us through colonial education, which was 100% church education when it first came to India in the 19th c.: not only mission schools, but all early Western universities such as Oxford and Cambridge were created by the church, and remained fully under its control till then.
Second, “Euclid’s” supposedly “superior” way of doing geometry was taught as part of the church curriculum for centuries. Why? That curriculum was designed to create missionaries. Future missionaries were taught the ability to persuade others: they were taught to use reason to persuade those who rejected the Christian scriptures. Hence, the church used math to teach reasoning, not practical calculation.
The third and least known fact is this: the word “reason” involves a tricky double speak. It does NOT refer to ordinary ways of reasoning, as people are easily tricked into believing. Rather it refers to a special way of reasoning developed by the church to support its “theology of reason” (which it adopted during the Crusades). Briefly, the church divorced reason from empirical facts. It had good reason to do so. Empirical facts are contrary to church dogmas: the notions of God, heaven, hell, resurrection, virgin birth are all contrary to the empirical. To defend its anti-empirical dogmas, the church declared empirical proofs to be inferior. It declared that “pure deductive proofs” based on reason, but divorced from facts, are infallible and “superior”. This church doctrine of reason is exactly what our school texts promote through the story of “Euclid” and his “superior” deductive proofs. Incidentally, that story also serves to hide the relation to church dogma.
Since the church used the book Elements as a textbook, its author had to be theologically correct, and early Greeks were the only people whom the church acknowledged as its “friends”. Hence, the author of the book was declared to be an unknown early Greek. The church never appointed a black or woman as pope, and it would have egg all over its face if it acknowledged the true author of the Elements as a heretical black woman who was raped and brutally killed in a church, as I asserted in my censored article. Science 2.0 did change the title of my article to “Was Euclid a black woman”, but did not add the part about her being lynched for being heretical.
The church used the book Elements by grossly “reinterpreting” the original. It was falsely asserted that the book contained pure deductive proofs, aligned to the church theology of using reason divorced from the empirical. This assertion, repeated by our school texts, is brazenly contrary to facts. The actual fact is that the book Elements does NOT have a single such pure deductive proof from its very first proposition to the last. Ironically, this itself shows how terribly fallible deductive proofs are—for centuries, invalid deductive proofs were wrongly accepted as valid by ALL Western scholars. When this truth was inevitably acknowledged, at the beginning of the 20th c., a quick substitute had to be invented to save Western pride from crumbling at a time when it was at its zenith.
The substitute for “Euclidean” math, invented by the West at the turn of the 20th c., was the formal mathematics of Hilbert and Russell. Russell’s proof of 1+1=2 is so complicated because one is not allowed to point out empirically that one apple and one apple make two apples. Formal mathematics mimics church dogma; it prohibits the use of the empirical, on the belief that the prohibition of the empirical leads to some “superior” form of truth.
This belief is pure balderdash. In fact, reason divorced from facts can be used to prove any nonsense whatsoever. To show this, I gave the example of the horned rabbit in my censored article. (1) All animals have two horns. (2) A rabbit is an animal, therefore, (3) a rabbit has two horns. Of course the conclusion is nonsense, and so is the premise (1). But we know that only as an empirical fact; if all reference to empirical facts is prohibited we have no way of knowing the truth or falsehood of premise (1). As Russell put it, in formal math we “take any hypothesis that seems amusing, and deduce its consequences”,8 and I am distinctly amused by the hypothesis that all animals have two horns, and its deduced consequences for rabbits. It illustrates the conclusions based on pure deduction which the church glorifies as infallible.
Others used reasoning differently together with empirical proof. For example, in India all traditional schools of philosophy accepted the empirical (pratyaksa) as the first means of proof. This was also true of traditional Indian math (normal math) from the time of the sulba sutra–s.9
Now, indoctrinated colonised minds often conflate acceptance of empirical with rejection of reasoning. But that is not true: like science, most systems of Indian philosophy, and traditional Indian math, accepted both empirical proofs and reasoning. The only exception was the Lokayata, or people’s philosophers, who warned against inference not based on the empirical. Their example of wolf’s paws is similar to the example of the horned rabbit above: on seeing the pug marks of a wolf, people in a city inferred that a wolf was around. Actually the pug marks were made by a man to demonstrate the foolishness of inference not based on sound empirical facts.
But this foolish dogma that avoiding empirical facts leads to a higher form of truth is what we still teach today. Early in middle school, children are introduced to formal math and avoidance of empirical as follows: the NCERT class 6 text asserts that a geometric point is invisible. It adds that a point determines a location. At two recent workshops, I asked a number of school math teachers and students how do they know what location a point determines since the point is invisible. They had no answer. But they had the honesty to admit their ignorance, unlike colonised intellectuals and “experts” who will defend the doctrine of invisible points exactly like the courtiers who defended the emperor’s invisible new clothes.
Many people say that math is difficult because it is abstract. This is wrong. The word dog is an abstraction, for dogs are of varying sizes and shapes. But children have no difficulty in understanding the abstraction “dog”, for one can easily point to a dog. Likewise children have no difficulty in understanding the abstraction dot, though dots come in various sizes and shapes and colours. But an invisible point is NOT such an abstraction: for one cannot point to a point. Nor can one infer the existence of invisible points from other phenomena the way one can infer the existence of electrons from tracks in a bubble chamber or infer fire from smoke. A geometric point, as taught in school today, is thus pure metaphysics; it has no real existence. People regrettably confound church metaphysics about unreal things with abstraction.
Further, a line too is asserted to be invisible by the NCERT 6th standard text. So I asked teachers and student how they can verify the postulate that exactly one straight line passes through any two points. They again had no answer. I also showed them that any two real dots can be connected by multiple straight-looking lines, so the postulate is not based on experience but solely on Western authority.
The church strategy of teaching about non-existent things forces students to abandon commonsense and rely on Western authority. Ultimately the only “reason” given by colonial education for why 1+1=2 is that some Western authority like Peano approved it! Those who resist this teaching, and try to understand on their own, are the one’s who find math difficult and abandon it.
But the text does not stop with one absurdity. After three years of allowing this nonsense about invisible points to sink in to the child’s mind, the NCERT class 9 text introduces a further piece of nonsense. It says that a point cannot also be defined in other words. (Ditto for line and plane.) It explains this as follows. If one says “A point is that which has no part” then one is obliged to define “part” and so on, leading to an infinite regress.
Such an infinite regress does NOT arise in the case of a dog or dot, for one can simply point to several instances of dots, terminating the regress. The cause of the infinite regress is the desire to preach the church dogma of avoiding the empirical. This motive is hidden, and never made clear by the NCERT text. Even the direct connection to church dogma is obscured by false stories of “Euclid”.
The matters of “Euclid” and the exclusion of the empirical are relatively simple. But if colonised minds have not understood even these simple tricks in two centuries, they are unlikely to ever understand the more complex tricks involved. The church would have permanently duped them through “education”. Thus, once students are conditioned to regard mathematics as pure metaphysics, a metaphysics of infinity crops up at every level: there an infinity of points in a line, an infinity of lines in a plane and so on. I reiterate that this metaphysics of infinity has nil practical value: a computer cannot handle any metaphysics of infinity, but most practical applications of math, such as sending a rocket to Mars, are accomplished using computers.
It is a common error to believe there is a unique notion of infinity, but I will not go into details here of how a particular metaphysics of infinity in math (especially calculus) forces us to accept church dogmas of eternity as part of science.10 Thus, the metaphysics which Europeans added to the Indian calculus forces time in physics to be like a line, as posited by core (post-Nicene) church dogma. This is the magic by which the metaphysics in math determines scientific “truth”. It is hard to explain this to people ignorant of math who are duped into thinking the belief in “laws of nature” is about science, though it is clearly a church dogma advanced by Aquinas in Summa Theologica.
We should change the teaching of math, and teach normal math solely for its practical value. Certainly we should do this at school level, for the benefit of the millions of students who drop out. They have no need to learn about the doctrine of invisible points and lines. The aim of colonial education was to teach people subordination to Western authority, but I dream of a new generation of children freed from the shackles which tie the colonised minds of many of our “educationists”.
But the tragedy is that the system cannot be changed. The students cannot change it. Even if the text is wrong, they are compelled to recite it under threat of failure. The teachers cannot change it, they must teach from the text or they risk losing their jobs. The government—ministers and bureaucrats—won’t risk changing it for they are ignorant and fear ridicule and loss of power if something goes wrong. The “experts” have a vested interest in it, so they will not change it. They refuse even to discuss things publicly.
There is an additional layer of protection which preserves myths and dogmas in math. This church method of censorship comes naturally to the colonised minds (intellectuals, journalists and so forth) who have been indoctrinated by the system. Unable to address an iota of the substantive critique, they will abuse, ridicule and reject and censor it. This censorship is done by loyal gatekeepers ignorant of both the philosophy of math and church theology. In this country of the blind, the two-eyed man is blinded because the colonially educated intelligentsia has been taught numerous subtle superstitions far more dangerous than astrology.
This is how the church controls mass behaviour—through mass superstitions—which it has smuggled into math and science to make those superstitions credible. Censorship defends that strategy. The only hope is that, today, people in the slums of Soweto understand what the colonised intellectuals elsewhere do not, that colonial education spreads superstitions packaged as part of math and science.
2C. K. Raju, Euclid and Jesus: How and why the church changed mathematics and Christianity across two religious wars, Multiversity and Citizens International, Penang, 2012.
3C. K. Raju, Cultural Foundations of Mathematics, Pearson Logman, 2007.
7C. K. Raju, “Black thoughts matter: decolonized math, academic censorship, and the ‘Pythagorean’ proposition”, Journal of Black Studies 48(3) April 2017, pp. 256-278. http://journals.sagepub.com/doi/abs/10.1177/0021934716688311.
8Bertrand Russell, “Mathematics and the metaphysicians”, in Mysticism and logic and other essay, Longmans, Green and Co., London, 1919, p. 75.
9C. K. Raju, “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhâsâ”, Philosophy East and West, 51(3), 2001, pp. 325–362. http://ckraju.net/papers/Hawaii.pdf.
10Those interested my check out the following. (a) “Eternity and Infinity: the Western misunderstanding of Indian mathematics and its consequences for science today.” American Philosophical Association Newsletter on Asian and Asian American Philosophers and Philosophies 14(2) (2015) pp. 27-33. Draft at http://ckraju.net/papers/Eternity-and-infinity.pdf. (b) The video of my Berlin talk, posted at https://youtu.be/eZ3SDf6u_DA. (c) The video of my MIT talk posted at https://youtu.be/IaodCGDjqzs, and the references in the abstract at http://ckraju.net/papers/Calculus-story-abstract.html.
C. K. Raju has authored several books, proposing a tilt in the arrow of time, and a new theory of gravitation in physics, and zeroism in math. He was the first to show that calculus developed in India and was transmitted to Europe in the 16th c. where it was misunderstood.