Structural Superstitions in Science

Continued from Previous Page

Abstract. Apart from Newton's superstitions, there are structural superstitions in science. Science is a social enterprise, which today relies heavily on the social criterion of reputability, not Popper's criterion of refutability. Colonial mathematics and the church superstitions (infallibility of deduction) on which it is based, provide a systematic way for the entry of church superstitions into science. Europe got most of current useful school mathematics from India. However, Europe got a religious geometry of mathesis from Egypt which was later reinterpreted and used for the church theology of reason during the Crusades. This theology is used to assert the "superiority" of colonial mathematics today. However, Indian ganita too used reason, though it used reasoning based on facts. Basically two type of reasoning are confounded: faith-based reasoning used by the church and fact-based reasoning used in Indian ganita, by the trick of using single word reason for both. Axiomatic reasoning is falsely attributed to Greeks, though it is completely absent in the "Euclid" book (whether or not "Euclid" existed). However, faith-based or axiomatic reasoning is very much found in Crusading church theology which could not accept facts which were all contrary to church dogma. Axiomatic proofs are decidedly NOT superior for any practical applications of math. But since mathematical axioms are ALL laid down by the West, and axiomatic mathematics is a consequence of and varies with those axioms, this way of doing math allows the West to control mathematical knowledge and through it science, as in singularity theory.

Structural superstitions in science

We saw in part 1 that colonial temper has consistently used a false history of science to lie about  and run down non-Western achievements in science. Equally, it has used a false history of science to boast of Western superiority, a claim directly linked to the earlier claims of Christian, then White superiority. Further, even the most prominent scientists in the West have superstitions. Belief in laws of nature is one superstition of scientists. Therefore, scientific temper does not equal imitation of what the science community does.

But is the argument limited to history of science or to terminology? No!  Church superstitions have penetrated the very fabric of present-day science. This also shows the extreme irresponsibility of scientists who today lament the lack of scientific temper: that they are lost in the colonial narrative and unwilling or unable to do anything to rectify the situation even as regards superstitions IN science.

The criterion of rePutability

To begin with a simple initial example, scientists today superstitiously believe secretive peer review,  a method invented by the church to preserve falsehoods, is the best guide to truth,[1] as opposed to the traditional Indian system of transparent and open public debate. Science today, like the West, heavily relies on authority, especially in physics and mathematics.[2]

This principle of secretive refereeing also serves tohide dissent in the manner of the church. My paper on escaping Western superstitions has remained unpublished for a decade due to the Western academic system of suppressing uncomfortable truths by pre-censorship,[3] and post-censorship,[4] and could be freely articulated only among Blacks.

Briefly, much physics today is based on social rePutability, NOT empirical refutability. For example the recent Nobel prize was given to Roger Penrose,[5] for a rePutable claim about black holes which is NOT empirically refutable. But all this is to be set aside because the real aim is the coarse political one of obtaining a political advantage by lies reiterating the superiority of the West and the inferiority of the non-West. This enables superstitions to be preserved in mathematics and science.

Colonial mathematics and its church superstitions

How Europe got most mathematics from India

Thus, mathematics is essential to science. Recall that the West, starting from Greeks and Romans, was very backward even in elementary arithmetic, hence imported arithmetic (“Arabic numerals), algebra, trigonometry and calculus[6], and probability[7] from India. Amazingly, despite persistent Western boasts of superiority, Westerners were so intellectually backward that they took a thousand years just to understand the elementary arithmetic they imported from India. The problems they had with zero (clear from its very name from sifr=cipher=mysterious code) are well known.

However, from the 12th century Fibonacci to the 19th c. de Morgan, Westerners were even more confused about negative numbers, with de Morgan a professor of math but, accustomed to primitive pebble arithmetic, declaring them to be impossible in his textbook on algebra[8] declared a prelude to calculus. It is so absurd that a prominent professor of the University College London declared negative numbers as impossible.

Asserting the superiority of colonial mathematics

Worse colonial education brought in a new type of mathematics, different from, say, traditional Indian ganita.[9] The essential new thing of colonial mathematics is—you guessed it—the assertion of Western superiority and the inferiority of all others. As we saw, this claim of Western superiority is directly related to earlier claims of Christian or White superiority. But unlike those earlier claims of Christian or White superiority, which are nowadays rarely or circumspectly publicly articulated, this assertion of Western superiority, which serves a similar purpose, is right there in our school texts, taught as part of a subject compulsory for all: 9th standard mathematics.

Thus, the NCERT class IX math text explicitly says that “Greeks” did a “superior” geometry compared to all others in the world (Babylonians, Egyptians, Indian) who supposedly did inferior geometry (since it was practical)

“Also, we find that in some civilizations like Babylonia, geometry remained a very practical oriented discipline, as was the case in India and Rome. The geometry developed by Egyptians mainly consisted of the statements of results. There were no general rules of the procedure. In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning (see Appendix 1).”[10]

It goes on to add the falsehood that all others in the world were also inferior because they had no notion of proof in mathematics:

“While mathematics was central to many ancient civilizations like Mesopotamia, Egypt, China and India, there is no clear evidence that they used proofs...”[11]

The obvious aim is to used mathematics to teach students that the West was superior and all others were inferior, in line with traditional church propaganda. But this is bald deception because the above quote from the school text is blatantly false in THREE different ways.

Mathematics, rationality and the two meanings of reason

The first is a brazen lie that others lacked reason/proof in mathematics. Indians did use both reason and proofs in mathematics. For example, the Indian Nyaya sutra,[12] verse 2, explicitly lists four means of proof (no such clear definition of proof is found in any Greek text).

It lists empirically manifest, deductive inference, analogy, and testimony as the means of proof. The second anumana involves deductive inference by reasoning. There are many examples of how reasoning was used in Indian mathematics (ganita) summarised in this chart:[13] for example Aryabhata reasoned that the earth is round from the observed fact that far-off trees cannot be seen:[14] he inferred it, and did not travel to space to see whether the earth is round. Therefore, the textbook claim that no one except the Greeks used reasoning or had proofs in mathematics is blatant false, and meant to instill colonial temper in the gullible.

The second falsehood in the textbook claim is subtler and more dangerous. The text book uses one word “reason” to deliberately and deceptively confound two distinct meanings and usages of “reason”. (As we will see, this confusing way of of using the word “reason” was a deliberate trick used by the Crusading church, and copied by colonial/church education.) Thus, as in the above example from Aryabhata, reasoning in India began with facts or observation: the empirically manifest was the first means of proof on which inferences were based. This is also the way reasoning is (or ought to be) used in science.

But that is NOT the way that reasoning is used in current/colonial mathematics, or to be precise, in formal or axiomatic mathematics. Formal mathematics prohibits the empirical.[15] This is clear enough to anyone who is mathematically literate enough to know the current definition of a mathematical proof.[16]But the prohibition of the empirical is also stated explicitly in the NCERT IX math text by telling the students to “Beware of being deceived by what you see”. (And blindly trust what you are told; that’s the beginning of scientific temper!?)

However, each statement in the proof has to be established using only logic.... Beware of being deceived by what you see (remember Fig A1.3)[17]

In other words, the truly novel feature of axiomatic or formal math is the prohibition of the empirical in axiomatic mathematical proof. The school text deceives children by asserting the novelty of reason when the real novelty is not the use of reasoning (which existed from long before the “Greeks”) but the EXCLUSION of facts. Instead of reasoning from facts, the novelty is to reason from axioms (=postulates).

Many people are further confused about the meaning of the word “axiom”; it does NOT mean a self-evident truth (no longer superstitiously assumed to exist): an axiom is the same thing as an assumption or postulate. As even the same class IX school text also admits (p. 83).

Now-a-days, "postulates" and "axioms" are terms that are used interchangeably and in the same sense.[18]

To summarise, there are two kinds of reasoning: (1) reasoning with facts and (2) reasoning without facts which prohibits the use of the empirical. The school text hides this novel way of reasoning without facts, by using only one word reason for both kinds of reasoning. People assume it means the first kind of reasoning with facts, when the school text actually means reasoning without facts, from axioms or assumptions.

That is the kind of reasoning, used in axiomatic mathematical proofs, which are being declared “superior” by the school text. In fact, the use of reasoning for proofs is found also in other cultures though the school text lies about it. What the school text should emphasize is novelty of the prohibition of facts. (Prohibiting facts a key part of scientific temper?) But the real aim of the school text is only to use mathematics to assert Western superiority, by lying and confusing young children using (as we will see) church propaganda about reason.

How does the prohibition of facts make a method of proof “superior”? It is one thing that the Vota of scientific temper persistently dodge facts. It is another that this exclusion affects should be made a principle of the mathematics on which science is based. So what is this peculiar kind of mathematics and science?

Lack of axiomatic proofs among Greeks

But before coming to that, let us first take up the third blatant lie in the school text. This is the claim that some Greeks used axiomatic reasoning in mathematics.

Recall that, in the context of the recent debate on removing the term “Pythagorean theorem” from our school texts, I raised an issue, challenging anyone to provide any axiomatic proof of the “Pythagorean proposition” before David Hilbert.[19] Merely citing the authority of some Westerner who said “Greeks had axiomatic proofs” won’t do because they are full of lies: an actual example is needed.

Thus, this falsehood was believed for centuries in the West, that the “Euclid” book (whosoever wrote it) had axiomatic proofs (though anyone who reads it cans see it has none, not even of its first proposition[20]). Alas, the superstitious Westerner for whom the “Euclid” text was prescribed reading for centuries, lacked the scientific temper to see that. That false belief of axiomatic proofs in the “Euclid” book makes the exact order of theorems in it important: thus the “Pythagorean theorem” is the second-last proposition in the “Euclid” book, not the first, as it is in Indian texts such as the Yuktibhasha[21] which allow use of empirical methods of proof. Cambridge University even foolishly changed its exam rules around 1880’s to incorporate this belief that the order of the theorems in the “Euclid” book was sacrosanct.[22]

Sadly for the credibility of Western authority, so important for our scientists and votaries of scientific temper, by 1900 it was publicly admitted that there were no axiomatic proofs in the “Euclid” book, as Bertrand Russell[23] explained. David Hilbert[24] rewrote the whole book to provide axiomatic proofs missing in it. Those with colonial temper, of course, think it is right and proper to hide this century-old fact (no axiomatic proofs in Euclid book) from children, to fool them about Greeks, for this fact about “Euclid” reflects poorly on Western intellects and credibility. To reiterate, as in my tweeted challenge, there are no axiomatic proofs in any Greek or Western work prior to Hilbert’s 1899 book. But the school text, talks of Greeks and does not admit this.

What is interesting is this: none of the colonised or the votaries of scientific temper will ever come forward and ask the school text to admit this. They don’t regard evidence as necessary for Western history and will not read and apply their mind to even the first proposition of the “authoritative” version of the “Euclid” book,[25] even if it is put before them. Alas that aspect of colonial/church temper is in the mathematics used in the very fabric of present-day science.

Axiomatic proofs in church theology

Why call it church temper? Because though axiomatic proofs (proofs which use reasoning minus facts) are NOT found in any Greek texts they ARE found in the Crusading church theology of reason. Greeks were once again just a proxy for the church, but this time to bolster its theology of reason now a part of the philosophy of mathematics. For example, Aquinas proved his angel theorem by reasoning from the axiom that angels occupy no space. There is a very good reason why the church prohibited facts because facts are contrary to most church dogmas. Obviously. there are no facts about angels, just a silly superstition. Because facts were so embarrassing, the church, when it adopted reason during the Crusades, prohibited facts. And that prohibition of facts is part of the mathematics used for science today!

Let us also understand why the church adopted reason during the Crusades. The Crusades were launched to convert Muslims by force, and thus grab their wealth, the way pagans were earlier converted by force. But the Crusades were sustained military failures: Muslims were militarily just too strong then. Therefore, the church turned back to persuasion. But at that time, the Bible was its only means of persuasion, and the Muslims rightly rejected the Bible as corrupted.

However, the information brought back by Crusading spies like Adelard of Bath (who first brought the “Euclid” book to the Christian world) was that Muslims accepted reason rather than authority.[26] Therefore, the church, led by Aquinas and his schoolmen adopted “universal reason”.

But the catch was this: the church accepted reason but could not very well also accepts facts, for all its dogmas were about imaginary things with no factual existence, like God, heaven, hell and angels etc. Aquinas’ insight was this: only facts went against church dogmas, not reason alone. Therefore, as Aquinas realized, the church could accept reason without facts, as in Aquinas’ theorem about angels.

Of course, right since Plato, the West, which copied Egyptian mystery geometry, has linked mathematics to religious beliefs.[27] Indeed, Plato connects mathematics to mathesis or soul arousal (today masked as aesthetics) and deprecated the practical applications of mathematics. Since this “pagan” not of soul was cursed by the church, the church has hence initially banned mathematics. But it later accepted mathematics during the Crusades. But while Plato deprecated the empirical. The church prohibited it.

Having adopted mathematics as reasoning without facts, the church declared it as
“superior” for the has always asserted that whatever it did was “superior”, as in the earlier assertions of Christian superiority (for genocide+slavery) then White superiority (for slavery) then Western superiority (for colonialism).

The point here is this: the most prominent Western intellects were totally hegemonised by the church and believed all these superstitious claims of superiority. Hence, the top Western minds like Hilbert and (racist) Russell were fooled into believing that axiomatic proofs were indeed superior so that the “Euclid” book (and all mathematics) should be redone using axiomatic proofs.[28] Both Hilbert and Russell were victims of this church superstition and hence rewrote mathematics using axiomatic proofs which they regarded as superior. The rest of the West followed them.

To summarise, there are NO axiomatic proofs in any Greek books, but they do exist in the theology of the Crusading church which falsely attributed the origin of such proofs to the Greeks, to dodge the charge of having invented a politically convenient theology. Due to church hegemony (through control of education in the West) the best Western minds foolishly believed in the purported superiority of such proofs based on reasoning without facts.

Are axiomatic proofs superior?

Finally, let us turn to the claim that axiomatic proofs are superior. Apart from church politics is there any other reason to believe that axiomatic proofs are superior to scientific proofs based on reasoning from facts? The proponents of scientific temper never argued that case for superiority.

For example, the kindergarten proof of 1+1=2 involves showing that 1 orange+1 orange make 2 oranges.[29] This is empirically manifest (pratyaksh).

However, since the empirical is prohibited in axiomatic math, Bertrand Russell in his Principia Mathematica needed 378 pages[30] to prove 1+1=2.

Why on earth is this superior? Just because of its extreme prolixity? Most people don’t understand the proof or anything on that page 378. It is also clear enough that the prolix proof does not offer any novel practical advantage, and ignorance of Russell’s method did not affect any practical thing: people have managed arithmetic wonderfully well for thousands of years before Russell.

So, what is superior about prohibiting facts? A common but deceptive argument is that empirical proofs are fallible. That is true, but no one yet abandoned science because of the existence of experimental error. The superstition is in the unstated part of the claim that prohibiting the empirical results in infallibility, which is a superstition exactly like the church’s clownish claim on the infallibility of the pope. But this is a superstition our proponents of scientific temper will doggedly refuse to admit.

Now, in India, the Lokayata long rejected deduction on the grounds that it is fallible and does not lead to the truth, for the starting assumptions (the axioms in this case) may be faulty. The Western philosophical counter to this is that deductively proved theorems are certain relative truths. But this counter is faulty, for deduction is never certain. There can be many errors in process of deduction[31] (not only in the axioms), especially in a complex proof. This is quite obvious from the many faulty proofs given by students which every math teacher has to correct. It is also clear from the many faulty proofs which are published even by authoritative mathematicians. So, publication is no proof of infallibility!

Indeed, given the fact that there are some errors in deductive proofs, how exactly does one know that there are no errors in Bertrand Russell’s 378 page proof of 1+1=2? Either one blindly trusts his authority (scientific temper??), or one checks it oneself. But checking a deductive proof is an inductive process: even if one has checked it a thousand times (very unlikely) there is still some philosophical doubt about an error in it, rather more serious doubt than that which purportedly persists for a snake mistaken for a rope, well after the snake is beaten to death.

In fact, for most people there is no question of checking Russell’s proof of 1+1=2, they simply don’t know the math. The process gets more complicated if one uses the real number 1, instead of 1 as a cardinal in Russell’s proof. Thus, in axiomatic math, the natural number 1 is NOT the same as the real number 1 which requires different axioms, the axioms of set theory, not Peano’s axioms. Since real numbers are taught in the first chapter of the same class IX school text, everyone ought to know about them, at least why1+1=2 in reals. Unfortunately, almost no one does. Hence, no one in JNU accepted my Rs 10 lakh Cape Town challenge to prove 1+1=2 in reals, from first principles, without assuming any theorems of set theory. Such a proof has never been attempted, to my knowledge, and may take a 1000 pages. How, then can they be sure that deductive proof are infallible! Obviously, they just blindly trust mathematical authority, and pass off this superstitious trust as scientific temper. Fear of mathematics facilitates that blind trust.

That brings us to a key political reason (apart from church superstitions) why the West promotes axiomatic mathematics today.[32] Axiomatic math enables the West to control mathematical knowledge. Since the axioms are all metaphysics (they are irrefutable) there is no way to verify them except by relying on authority. This was originally a church trick of using metaphysics (in e.g., ethics) to enforce reliance on its authority.

After colonialism, authority resides in the West, and forcing people to rely on authority rather than their sense data is a way to enforce the colonial temper of “trust the West” even for 1+1=2, “don’t believe your senses” as the textbook says. Because colonialism centralized all authority in the West, all the axioms of axiomatic mathematics were actually laid down by Westerners. Axioms decide theorems, and mathematical knowledge according to axiomatic math. Hence, axiomatic math enables the West to control mathematical knowledge, hence all knowledge which depends on it.

To explain the metaphysics in axiomatic math, consider, for example, the simplest axiom that there is a unique straight line through two points. As our NCERT class VI school text explains points must be invisible! Unlike invisible electrons, geometric points have no indirect consequences, so declaring points as invisible leaves us no empirical way to know where exactly a point is. Lines made up of invisible points are naturally invisible, and, obviously, there is no way to verify or refute that a unique invisible straight line passes through two invisible points. On the other hand, if points are of dots of finite size, hence visible, as invariably happens in practice, there is always a way to connect two points with multiple straight lines,[33] so that the axiom is false.

At a more advanced level of, say, “real” numbers, one needs a metaphysics of infinity which is closely allied to church dogmas of eternity, as I have explained in detail elsewhere.[34] It is hard to explain this, since most people do not understand that set theory is a metaphysics of infinity. Perhaps the simplest way to understand this is that (a) the use of calculus for physics together with the (b) the superstitious belief that axiomatic real numbers are needed for calculus, forces the conclusion (c) that time in physics must be like a straight line. Note how this conclusion about the physical nature of time was pushed into physics through metaphysical axioms of mathematics. How this way of looking at time has a religious bias (e.g., it is pro-Christian and anti-Hindu), requires a whole book. Since I already wrote such a book long ago,[35] there is no need to go into further details here. Similar biases can also be pushed from axiomatic math not only into science but also into economics for example (Arrow’s impossibility theorem).

To summarise, axiomatic reasoning originated with the church, not with “Euclid” or any Greeks. But our school texts systematically deceive school children about it, and NCERT and the votaries of scientific temper refuse to deal with the historical evidence. The purported superiority of axiomatic proof (reasoning without facts) is a mere church superstition due to the political exigencies of the Crusading church. However, this church superstition still provides the West with leverage to control mathematical knowledge hence, indirectly, scientific knowledge. Today, people well understand the economic and political implications of controlling knowledge: as in the Nobel prize given for Arrow’s nonsense impossibility theorem. This is similar to the way earlier immensely evil church superstitions about Christian and then White superiority provided the West immense economic benefits by declaring vast amounts of land and African slaves as Christian/White property, except that this new superstition operates in context of a new economy where knowledge is a key means of production of wealth.

Axiomatic mathematics is just a superstitious principle of Western bhakti in mathematics. It is useful only to the extent that it mimics and reproduces the results of normal mathematics, much if it including efficient arithmetic, algebra, calculus, trigonometry and probability from India.[36] Our proponents of scientific temper, such as Jayant Narlikar (an author of the NCERT text) are dead silent[37] on the lack of scientific temper in axiomatic mathematics, used by scientists worldwide. A former Director of NCERT, Parvin Sinclair,[38] a mathematician, refuses to discuss the nath text she wrote and authorised.

The unstated aim of these politicians masquerading as academics is to corrupt the minds of millions of children by implanting such Western superstitions. Our college math teachers, such as Nandita Narain[39] of Stephen’s college are equally intellectually dishonest and refuse to discuss publicly the trash metaphysics of real numbers which they teach in the math classroom. That’s colonial temper: believe the false history of Greeks (a proxy for the church/Whites) without evidence and on faith in the West. Teach a bad (church) philosophy of axiomatic math in school texts and in the math classroom, and then dodge all intellectual challenges and responsibilities regarding it to save the faith in Western superiority and damn the colonial children who are victims.

Summary and Conclusions

Because the Western practice of science developed under the hegemony of the church, there still are many structural superstitions in the Western practice of science today. This starts from the criterion of rePutability, instead of refutability, used to decide the validity of scientific theories.

However, the most important superstition in science concerns colonial or formal mathematics. Though the West was arithmetically challenged and took a 1000 years to learn even basic arithmetic from India, it makes the stock claim of superiority. This “superiority” is based on the superstitious belief that reasoning without facts, as the church did, is a superior way of reasoning. As I have repeatedly pointed out all practical value of mathematics for science, from arithmetic to calculus, still comes from the normal mathematics that Europe imported. Obviously, metaphysics of infinity which the West injected cannot be used for practical calculation.

The claim of the superiority of axiomatic math is just another superstition which is politically useful, since it helps the West to dominate mathematical knowledge, and through that all knowledge.

Our votaries of scientific temper never bother to challenge this church superstition in mathematics, which is part of science. The alternative is that they should defend that claim of superiority publicly, which they cannot since it is as hard to defend as the earlier superstitious claim of White or Christian superiority. So, they just sit mum.

That brings us to the ultimate question: what right do those scientists have to talk of lack of scientific temper when they can’t have scientific temper even as regards the content of science? When they do nothing to eliminate the church superstitions in mathematics (axiomatic mathematics) in the structure of present-day science? And do not intervene in the superstitions taught to millions of school children by colonial education to instill a sense of inferiority in them. (Exception: Gauhar Raza, part of the conference, at least signed my 2014 petition to teach religiously neutral mathematics.[40]) Clearly, this is a case of the Biblical beam in the eye of the scientist which the scientist dishonestly refuses to address for his economic benefit.

The colonized need to understand they should not trust these claims by Westerners who have long been using such superstition of superiority to gain numerous advantages such as appropriating land, labour and knowledge. Since the claim of superior axiomatic math which is actually inferior cannot or will not be defended in public debate, and offers no superior practical value, the superstition must be abandoned. Especially don’t expect our university mathematicians, trained exclusively in Western mathematics, to offer a solution, or to validate one when offered.

Footnotes

[1] C. K. Raju, Ending Academic Imperialism (Multiversity, 2011).
[2] “Indic thought and contemporary science”, video , summary.
[3] “Western superstitions packaged as science”, meeting in Soweto.
[4] C. K. Raju, Mathematics, Decolonisation and Censorship, 2017.
[5] C. K. Raju, ‘A Singular Nobel?’, Mainstream 59, no. 7 (30 January 2021).
[6] C. K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE (Pearson Longman, 2007).
[7] C. K. Raju, ‘Probability in Ancient India’, in Handbook of Philosophy of Statistics, ed. Paul Thagard Dov M. Gabbay and John Woods, vol. 7, Handbook of Philosophy of Science (Elsevier, 2011), 1175–96.
[8] Augustus de Morgan, Elements of Algebra: Preliminary to the Differential Calculus, 2nd ed. (London: Taylor and Walton, 1837).
[9] C. K. Raju, 'Ganita vs Mathematics’, Himanjali 20, no. July-December (2020): 34–44. Or see this more recent video
[10] NCERT, Class IX, Math, p. 79.
[11] NCERT, Class IX, Math, p. 287, Appendix 1
[12] Satish Chandra Vidyabhushana, The Nyaya Sutras of Gotama (Allahabad: P?nin? Office, 1913).
[13] http://ckraju.net/papers/presentations/images/Proof-table.html.
[14] Note that the uses of reasoning in India long predates any notional date that may be assigned to the Nyaya sutra: the Lokayata rejected anumana, while Buddhists accepted it. In particular, the use of reasoning in India long pre-dates “Aristotle” (both the concocted Aristotle of Toledo, and the historical Aristotle of Stagira). Europe first obtained reasoning and “Aristotle” during the Crusades from Arabs (Ibn Rushd), and Arabs probably got it from India. Briefly the “Aristotelian” syllogism, not found in Greece is very similar to the Nyaya syllogism, and copied from it, sice the Indian one is earlier. C. K. Raju, ‘Logic’, in Encyclopedia of Non-Western Science, Technology and Medicine, ed. Helaine Selin (Dordrecht: Springer, 2016 2008).
[15] C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibh???’, Philosophy East and West 51, no. 3 (2001): 325–62.
[16] E.g. L. Mendelson, Introduction to Mathematical Logic (New York: van Nostrand Reinhold, 1964).
[17] NCERT Class IX, Math, Appendix 1, p. 301, emphasis original.
[18] But, then, to hide the significance of this admission for the understanding of mathematics, the text immediately goes on to the completely irrelevant issue of grammar: "postulate" is actually a verb.
[19] See point (3) in this tweet.
[20] C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and the Rant of Racist and Civilizational Superiority - Part 2’, Arumaruka: Journal of Conversational Thinking 1, no. 2 (2021): 57–105.
[21] Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibh???’.
[22] http://ckraju.net/geometry/cambridge-note.html.
[23] B. Russell, ‘The Teaching of Euclid’, The Mathematical Gazette 2, no. 33 (1902): 165–67.
[24] David Hilbert, The Foundations of Geometry (The Open Court Publishing Co., La Salle, 1950).
[25] T. L. Heath, The Thirteen Books of Euclid’s Elements (New York: Dover Publications, 1956).
[26] C. K. Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars (Penang: Multiversity and Citizens International, 2012).
[27] C. K. Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars (Penang: Multiversity and Citizens International, 2012).
[28] It doesn’t help to say that Russell was against the church—he was an agnostic, As I have earlier explained it is mere propaganda to claim that Stephen Hawking was an atheist so he did not mean to take advantage of the creationist conclusions from singularity theory. C. K. Raju, ‘A Singular Nobel?’, Mainstream 59, no. 7 (30 January 2021).
[29] For that typical kindergarten image, see http://ckraju.net/papers/presentations/images/oneplusone.jpg.
[30] For an image of that page 378, see http://ckraju.net/papers/presentations/images/Principia-p378.jpg.
[31] C. K. Raju, ‘Decolonising Mathematics’, AlterNation 25, no. 2 (2018): 12–43b.
[32] C. K. Raju, ‘Why axiomatic math is racist’.
[33] For an image of multiple straight lines connecting two dots, see http://ckraju.net/papers/presentations/images/two-dots.png.
[34] C. K. Raju, ‘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, no. 2 (2015): 27-33.
[35] C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003).
[36] C. K. Raju, ‘Precolonial Appropriations of Indian Ganita: Epistemic Issues’ (International round table on Indology, IIAS, Shimla, 2020).
[37] Reminder to J. V. Narlikar for a public discussion on the NCERT school math text for which he was lead author.
[38] Invite to Parvin Sinclair, author of the NCERT school math text, and former Director of NCERT.
[39] Invite to Nandita Narain to publicly debate her teaching of real analysis in the math classroom.
[40] https://www.ipetitions.com/petition/teach-religiously-neutral-mathematics. See, also, C. K. Raju, ‘The Religious Roots of Mathematics’, Theory, Culture & Society 23 (March 2006): 95–97; C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 1: Formal Mathematics as Biased Metaphysics’, Science and Culture 77, no. 7–8 (2011): 274–79. arXiv:1312.2099.