The previous article[1] explained why the terminology of the “Pythagorean theorem” is racist. That terminology is defended by appealing to the two myths of “Euclid”, used to assert that White Greeks did “superior” geometry compared to Black Egyptians. But both myths of “Euclid” (the myth of the person “Euclid”, and the myth about the book “Euclid” that it has axiomatic proofs in it) are false.
It is a completely bogus claim that “Greeks used axiomatic proofs which originated with Euclid”. There are no axiomatic proofs in the actual “Euclid” book, whoever wrote it. This fact was publicly admitted by Bertrand Russell[2] and David Hilbert[3] over a century ago.
How then did axiomatic proofs actually originate? Who FIRST advocated axiomatic proofs? These questions are important, for axiomatic proofs are what are used in the math taught in universities and schools today. But Western mathematicians (and historians) never asked this question about the real origin of axiomatic proofs. All Westerners gullibly believed for centuries in the myth of axiomatic proofs in the “Euclid” book. Today they just carry on believing in the Greek-birth myth of axiomatic math (like the virgin-birth myth, underlying church ethics), using myth jumping, or the tactic of defending one myth by invoking another myth as evidence.
But if the “Euclid” book as the origin of axiomatic proofs is just a false creation myth, one has to ask this question about the real origins of axiomatic proof, and answer it.
Why the Crusading Church adopted reason
The fact is that the earliest recorded case of actually (not mythically) using the axiomatic method of proof is its use by the church. An influential priest of the Crusading church, Thomas Aquinas,[4] used axiomatic reasoning to prove his “angel theorem” that many angels can fit on a pin. That is, axiomatic proofs originated during the Crusades, shortly after the “Euclid” book arrived in Europe as a crusading trophy. This correlation was later confounded with a causal relation, that “axiomatic proofs originated with Euclid”!
But why did Aquinas want to use reason? The context of the Crusades is critically important. Crusading spies like Adelard of Bath (who first brought the “Euclid” book from Muslim lands to Christian Europe) explained that Muslims accepted reason, unlike Christians who went by authority.[5] Indeed, there was a whole Islamic theology of reason, called the aql-i-kalam.[6]
This observation by the Crusading spy Adelard, that Muslims used reason, was very important for the real aim of the Crusades, which was to grab Muslim wealth, by trying to convert Muslims by the sword. But force failed against Muslims for centuries (after the first Crusade). The then military inadequacy of all the squabbling nations of Europe, combined under a religious flag, was amply manifest by the time of Aquinas. So, it was evident that Muslims could not be converted to Christianity by the sword, which method had earlier succeeded with pagans in Europe. This method (the sword) did not work even in the then Muslim parts of Europe (al Andalus, Spain etc.). These were the wealthiest parts of Europe then, hence a prime target of church greed.
Muslims obviously rejected the authority of the church. They also regarded the Bible as corrupted by the church, as even the scientist Isaac Newton did.[7] Force, church authority, and Bible all three were ineffective. But what else could work to convert Muslims? That is, the church had no option but to turn to reason to try to persuade Muslims to convert by reasoning. For this purpose, the church first erected a Christian theology of reason, parallel to the Islamic theology of reason. But how? The word “reason” occurs less than 100 times in the Bible, depending on the translation. So what were the church sources for this transformation? The church could hardly acknowledge that it was mimicking its religious enemy.
Therefore, the first step the church took was to appropriate reason and claim it as a Christian inheritance: this was achieved by its traditional secular method[8] of fooling opponents by concocting false history, since Orosius. False Crusading history simply attributed all “useful” knowledge in Arabic books (captured from Muslims) to early Greeks.[9] It was asserted that key Arabic books (incorporating world knowledge) were of Greek origin. The thought did not cross the minds of gullible Westerners that this tale of the sudden recovery of a long-lost heritage, during a religious war, might by a political stratagem.
Asserting the myth of the Greek origins of “reason”, in “Aristotle” and “Euclid”, enabled reason to be appropriated as a Christian inheritance, on the belief (since Eusebius) that the early Greeks were the sole “friends of Christians” (since they were before Christianity). So strong was the grip of the Crusading church on the minds of Westerners, that gullible Westerners are still sold on those Crusading myths of the Greek origins of reason. We have already examined, in the first part of the article, the falsehood of the myths about Euclid. (“Aristotle” will not concern us here.)
How and why the church invented axiomatic reasoning
However, even if reason came to be regarded as a Christian inheritance, it could still be dangerous to church dogmas. This danger was recognized, before reason was accepted into the church fold. Hence the Inquisition placed a number of reason-related books, translated from Arabic, on its notorious “Index”, and persecuted those engaging with these books.
This was the problem (resolving the danger of reason to church dogma) to which Aquinas and his schoolmen found a simple solution. The solution was to subtly twist reason to turn it into a church tool, so that “reason” could be used to support church dogma, and prove exactly what the church wanted: neither more nor less. How? By inventing axiomatic reasoning!
To understand how the church twisted reason, let us review the difference between normal reasoning and formal reasoning (or axiomatic reasoning). Normal reasoning begins with facts or observations. We see smoke, and reason that there must be fire. This is a good way of reasoning, and is very ancient;[10] its use in India is documented from long before any fictitious Greeks, such as Aristotle (of Toledo[11]) or “Euclid”. This is the kind of reasoning used also in science: for facts and observations are a key part of science.
However, Aquinas could not reason in this normal (or scientific) manner, for facts or observations are contrary to church dogmas about the nature of God [theology=God-knowledge]). Therefore, the church invented axiomatic reasoning, or reasoning which excludes facts.
To reiterate, the novel aspect of axiomatic reasoning is NOT the mere use of reasoning, which is ancient, but the prohibition of the empirical (facts, observations).
For example, Aquinas wanted to and did reason about angels. But who has observed angels? Obviously, no one. And what are the facts about angels? Obviously, none. Aquinas’ insight was that reason itself did not go against church dogmas, only facts did. Therefore, the church prohibited facts, and, as the starting point of reasoning, replaced facts by assumptions or axioms. Such axiomatic reasoning (= “reasoning MINUS facts”) was acceptable to the church, and it accepted it.
Therefore, Aquinas’ began his reasoning about angels not from facts, but from an assumption (or axiom) that angels occupy no space. [Note that “axiom” today means the same thing as “postulate” or assumption; specifically, it is NOT a “self-evident truth”, as some people wrongly imagine.[12]]
The other key fact about axiomatic reasoning (not so well known) is that axiomatic reasoning can be used to prove ANY pre-desired conclusion, by suitably tailoring the axioms. In actual practice, some subtlety may be needed to make the axioms seem plausible. But that plausibility, like the plausibility of a piece of fiction, is a matter of playing psychological tricks, not truth. That is, axiomatic reasoning made it easy to prove any desired “truth” using reason, in a way dependent on the church’s will.
To reiterate, by selecting assumptions appropriately, one can reason and arrive at ANY pre-desired conclusion whatsoever. This was true of Aquinas’ angel theorem, but it is equally true in axiomatic math today. By glorifying this sort of (axiomatic) reasoning, and declaring it “superior” the church could have its reason and keep its dogmas too!
“Euclid” as a mask for the church
To avoid any suspicion of deliberate intrigue, or any charge of tampering with the very notion of “reasoning”, the church concocted the “Euclid” myth. That is, the church claimed this method of axiomatic reasoning was ancient (for as Adelard also said, people, in his Crusading times, trusted ancient sources). Of course, we have seen that this claim, that axiomatic proof preceded the Crusades, was absolutely false: there are no axiomatic proofs in the “Euclid” book.
That is, having concocted the myth of person “Euclid”, as a theologically acceptable early Greek source, the church then easily “reinterpreted” the “Euclid” book and just falsely claimed the book had axiomatic proofs in it, in support of the method of reasoning (minus facts) politically useful to the church. The church has mastery in this art of “reinterpretation”, having repeatedly reinterpreted the Bible for centuries, to suit its immediate political convenience. This medievalism is exactly “postmodernism”: one can “interpret” a text any way one likes, irrespective of what is written in it, and irrespective of what the author intended: obviously a very convenient way to do “history” from textual sources! Of course, simpler tricks like forgeries continued to be used.
An example is the “curse of Ham” (more correctly the “curse of Kam”) used for the Bible defence of slavery,[13] since slavery was a source of much money. This was used to declare Blacks as inferior even after Black slaves converted to Christianity. (That church politics is the original source of colour prejudice.)
As we saw, the extreme gullibility of Western intellectuals, and their susceptibility to church propaganda, is on record: ALL Westerners believed for centuries this silly claim of axiomatic proofs in the “Euclid” book. Not one Westerner ever asked: what social circumstances in “Euclid’s” purported time compelled him to seek a special kind of proof not used before? What were the objections to the prevailing notion of proof in his purported times? What motivated him to select this particular kind of axiomatic proof, divorced from facts?
So gullible are Westerners, that in eight centuries not one Westerner had even the minimal scepticism to ask: how come the “Euclid” book was so divorced from the Platonic geometry (or Egyptian mystery geometry) prevalent among the early Greeks, and how come it exactly anticipated the political requirements of the Crusading church for axiomatic proof, so that the church used the “Euclid” book as a textbook for centuries? Believers in virgin birth are so full of faith and respect for church authority, they will believe anything.
And, today, because Western propaganda is all about faith in their authority, Western scholars have responded to my criticism ONLY by personal attacks based on slimy lies, or misrepresentations, and racist and colonial abuse. The response shows that the criticism bites, but they have no better defence. Indeed, on the ancient Indian method of reasoned debate, personal attacks and lies or misrepresentations are regarded as one of the 23 ways of losing an argument. One expects such responses only from lumpen elements.
But abuse and gross falsehoods have been the sole response from scholars from leading Western institutions such as Cambridge,[14] Oxford,[15] Harvard,[16] Cape Town;[17] that demonstrates the total intellectual bankruptcy of the West. The resort to blatant falsehoods is an indirect admission that Western scholars are incapable of dealing with the truth of the critique. The church never taught them that part! 😀
The other response has been censorship.[18] In twenty years, not one Western “scholar” could engage with my criticism, or seriously counter even a single point I made. Censorship is a convenient alternative to just delete the criticism.
Clearly, these are all propagandist responses unconcerned with truth, for example because they address the person not the argument. In fact, they involve a monkey-level[19] display of crude racist/colonial authority by “top” Western “scholars”. But such a persistently low-intellectual level, even of propaganda, from “leading” Western institutions, is just further proof of the intellectual inferiority of the West.[20] Obviously, any nonsense can be established as truth in this way.
“Euclid’s” intentions and diagrams
For the record, led us also address the apologia that the “Euclid book” intended to provide axiomatic proofs. Of course, it is hard enough to understand the intent of a real person, and anything at all can be claimed about the “intentions” of a non-existent person, but let us see the actual book.
Actually, the “Euclid” book does belong to the Platonic or Neoplatonic tradition of Egyptian mystery geometry. This is clear from the fact that the “Euclid” book is full of diagrams, and diagrams are a hallmark of Plato’s (Egyptian) mystery geometry of arousing the soul. Socrates in Meno[21] uses diagrams to arouse the slave boy’s soul and make him recall his innate knowledge of geometry. Plato’s Phaedo[22] explicitly asserted that diagrams help mathesis, and Proclus[23] hence cites Phaedo. And as Russell also explained,[24] diagrams are irrelevant (and misleading) for axiomatic proof.
But the church just could not accept this “Platonic” way of understanding the “Euclid” book, as related to soul arousal. Why not? Because it had pronounced its great curse (anathema)[25] on the related “pagan”/early Christian notion of soul. And there is this amazing concordance between the hidden political agenda of the church, and Western “authoritative opinion” of the “right” way to read “Euclid”! We will, in the next part of this article, see exactly how these superstitions have been tied to the self-interest of present-day formal mathematicians.
How the church SOLD its twisted method of reasoning
Of course, it was NOT enough to say that “axiomatic proof originated with “Euclid”. Why should (non-church) people use axiomatic proofs? To that end, the church used its traditional polemic of superiority: it said axiomatic reasoning is a “superior” way of reasoning (hiding the fact it was politically useful for church theology). It was on this ground of teaching “superior” methods that axiomatic math was globalised by colonialism.
But is this assertion (that axiomatic proof is “superior”) anything more than another silly church superstition, planted by the church, exactly like the church assertion (curse of Ham) that Whites are “superior” Christians? Let us examine it.
The practical inferiority of the axiomatic method
In what sense is axiomatic proof superior? Certainly, axiomatic proof does not add superior practical value: for example, the axiomatic proof of 1+ 1 = 2 does not add to the practical value of elementary arithmetic in a grocer’s shop.[26]
To be quite precise, traditional knowledge of arithmetic is useful in a grocer’s shop. If one buys a dozen objects in a grocery shop one should be able to correlate the number 12 with the observed number of objects. This involves an empirical process of observation. Hence, this is normal mathematics. Furthermore, one should be able to multiply to be able to calculate the price of a dozen objects, given the price of one. But all these processes were going on for thousands of years.
The question is what “superior” practical value did axiomatic proof (for example, from Peano’s 19th c. axioms) of 1+1=2 add to the process? And what “superior” practical value did Russell’s 378-page axiomatic proof of 1+1=2 add in a grocer’s shop? Obviously, nothing. Most people are completely unaware of Peano’s axioms or Russell’s complex proof of 1+1=2, and still manage to do their groceries splendidly. On the other hand, prohibiting the empirical makes it impossible to operate in a grocer’s shop. Expressing contempt for the practical value of math, as the formal mathematician so often does, is just an aggressive way to admit that formal math lacks practical value. One can more easily express contempt for the utter uselessness of formal math.
Some people worry, because they are ignorant of math. Grocery shop OK, they think, but what about the technologically advanced applications of math? Might we not lose something there? But, technologically sophisticated applications of math are no different: lack of practical value of axiomatic math is equally true in the matter of calculating rocket trajectories (or doing data science or machine learning). They are all done using normal math.
For example, rocket trajectories are calculated using calculus. On the axiomatic method, calculus requires real numbers,[27] as taught in all universities today. But actual calculations of rocket trajectories are done on computers which use floating-point numbers, and CANNOT use real numbers.[28] (In fact, real numbers cannot be used even if the calculations are done by hand, because even a single real number, such as π, can never be written down exactly by hand.) That is, for the practical value, one must reject the impractical metaphysics of infinity used for axiomatic real numbers, and come right back to the finite processes of normal mathematics. Likewise, machine learning, or data science uses statistics, which again needs calculus, but is done with floats on computers.
This practical rejection of axiomatic math, for normal math, is kept tacit. Even when explicitly acknowledged (as in “computational errors”) normal math is treated as erroneous and inferior. Because those ignorant of math confound normal and formal math, this helps to spread to spread the “it works” superstition among people, who fail to understand that what “works” in math is only normal math, not the metaphysics of formal math.
That is, despite the superstitious fears of the ignorant that “modern” axiomatic math might offer some practical value, which would be lost, it does not provide practical value. A computer cannot do real numbers, or measure theory or the Lebesgue integral or any other metaphysics of infinity which characterizes axiomatic math. What a computer does in statistics is normal math, and that sort of probability theory is very ancient.[29] (I repeatedly refer to the computer because a computer is tied to the real world; unlike humans, it cannot deceive itself that it “understands” a metaphysics of infinity.)
That is, formal math adds nothing of practical value to the applications of math, at any level of technology.
So, why is the normal and practical way of doing things declared “inferior”? Just to fortify the sense of superiority of Christians/Whites/Westerners who invented the axiomatic method?
Fallibility of axiomatic proofs and the purported rigor of axiomatic math
We saw that when their myths are challenged and hard evidence demanded, Westerners behave like myth jumpers: they hop from one myth to another treating further myths as “evidence” to support the first myth. Likewise, when cornered regarding the lack of practical value of formal math, church supporters and formal mathematicians will now jump to a different tack.
They will claim that regardless of practical value, axiomatic math is “rigorous”; they will say that is hence epistemically superior. They will say empirical proofs are fallible. The last sentence is true. But what is the problem with that? Science, too, uses fallible empirical proofs, but it is still the best means of knowledge that we have.
The real problem is with the unstated or understated part: that axiomatic proofs are “superior” because they are infallible. Is the belief in the infallibility of axiomatic (deductive) proofs anything more than a silly church superstition,[30] like the belief in the infallibility of the pope.
Because it sounds intrinsically silly to say that someone or something is infallible, this part is never clearly articulated. Therefore, that claim of infallibility of deduction needs to be put on the table under a bright light so that it is easily visible. How does the prohibition of facts result in infallibility? Obviously, if both (deduction and empirical proofs) are fallible, then one should only be comparing their degrees of fallibility and no Western philosopher ever did that.
Since I have discussed this issue, of the purported infallibility of deductive/axiomatic proof at length[31] and will just summarise.
Note, first, that deductive reasoning and axiomatic reasoning are often confounded, as in this example from an Indian school text.[32] Of course, an axiomatic proof uses deductive reasoning, so any weakness of deductive proofs applies also to axiomatic proofs. But the key difference is that axiomatic proofs, in present-day math, prohibit the empirical. So, let us start with deduction.
Where exactly is the infallibility of deduction? Setting aside simple 3 sentence syllogisms, which are fine for illustrations, real-life deductive proofs involve far greater complexities. Hence, students of mathematics (or simpler symbolic logic) make mistakes all the time, in following or reproducing such proofs. Therefore, they fail in math tests. Students are judged by authorities who ultimately decide whether a deductive proof by a student is right or wrong. But authorities, too, make mistakes, as in claiming wrong proofs of, say, the Riemann hypothesis many of which have been published.
If one does not rely blindly on authority, then given that mistakes in deductive proofs (or the deductive component of axiomatic proofs) do happen, a purported deductive/axiomatic proof needs to be repeatedly (inductively) re-checked by the person producing it.
While doing my PhD, I had some new junction conditions in general relativity[33] which involved a new mathematics (“products of Schwartz distributions”), hence had to be explicitly worked out. Those days, symbolic calculations were done by hand, and it took me over 100 sheets of paper. Naturally, I could not be sure of the validity of the novel conclusions,[34] and had to repeat the calculations afresh, stopping only when I got the same results twice. This took months.
That is, the validity of complex deductive proofs can only be decided either inductively (through a process of repeated rechecking by the author) or by blind trust in authority. Given that both induction and authority are both fallible, therefore, deductive proofs are at least as fallible as empirical proof. They must be re-checked inductively (and one can never be sure if the final answer is right).
In fact, deductive proofs are a lot more fallible; deduction is more frequently fallible. This is evident, for example, from the game of chess, which involves pure deduction. As Hardy[35] admitted “chess…is genuine mathematics”, meaning it is all about deduction. But every human being (including the ultimate human authority, the world chess champion) makes a mistake almost every time, hence loses to a machine (i.e., a computer program such as AlphaZero). (One can lose in the game of chess only if one makes a mistake; an error-free game must end in a draw.)
Therefore, if fallibility of proof is the deciding criterion, deductive proofs must certainly be rejected as inferior since more frequently fallible.
The epistemic weakness of axiomatic math
Proof (such as proof in normal math or proof in science, or in ancient India[36]) certainly has epistemic value. But that concerns proofs which accept the empirical, hence the real world. But the epistemic value of axiomatic proof, or proof which rejects the empirical, or avoids the real world, is just another church superstition.
The hilarious fact is that deductive proofs were rejected as epistemically inferior, by some sections of the non-West, long before the purported date of “Euclid”, or Aristotle (of Toledo or Stagira), and long before the Crusading church declared axiomatic/deductive proofs as “superior”. But this rejection of deductive proofs as inferior is no part of any Western discourse. It is never mentioned because all opposing viewpoints were blocked by the church, as the only way it had to establish its viewpoint about deduction, and that is tradition of blocking others is still the live tradition in the West.
But let us consider it. The argument, given by the Indian Lokayata (“people’s philosophers”), was sound: they rejected deduction[37] saying it does not lead to valid knowledge. For example, because one may start with wrong assumptions (or axioms).
Present day formal mathematicians cannot reject this argument. But they try to sugar-coat it by saying that the theorems of formal mathematics are “relative truths”, relative to the axioms. This hides the fact that any nonsense whatsoever may be a “relative truth”. (That is exactly why the Crusading church loved this method of reasoning and declared it “superior”.)
My rabbit theorem was intended to illustrate this point (“any nonsense whatsoever can be a relative truth”). “All animals have two horns, a rabbit is animal, therefore a rabbit has two horns.” This is an impeccable piece of deduction; and the conclusion “All rabbits have two horns” is a valid mathematical theorem. Of course, the theorem is obviously not valid knowledge. It is complete nonsense: that was exactly the point I was making: that a validly proved mathematical theorem need not be valid knowledge. I thought this was an extremely elementary aspect of axiomatic mathematics which I learnt long ago; as Russell put it, “in [axiomatic] mathematics one never knows whether what one is saying is true”.[38]
The key reason why the conclusion of the rabbit theorem is false (“relative truth”!) is that the axiom used, that “all animals have two horns”, is obviously false in the real world. However, the axiom is empirically false; a matter of everyday observation. But that observation is irrelevant to axiomatic math. As has been reiterated several times: axiomatic proofs are divorced from the empirical.[39] This divorce of the empirical from axiomatic proofs is also explained in the NCERT class IX math school text.
“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)!”[40] [Appendix 1, p. 301, emphasis original]
In other words, beware of judging the axiom “all animals have two horns” by what you see! 😄However, most people don’t know even their compulsory class 9 math properly. Or they don’t believe that axiomatic math can be so separate from our most basic means of knowledge: observation. Without the slightest understanding of axiomatic math, they sit in judgment on the critique of axiomatic math,[41] and therefore are easily and repeatedly fooled.
Axioms of math are metaphysics: evade the real world
Is there some way in which the theorems of axiomatic math can be really true (true in the real world not just “relatively true”)? For example, what if the axioms are empirically true? Alas, no. The exclusion of facts applies also to axioms or assumptions of math, the starting point of a proof. But this aspect of axiomatic math (the exclusion of the empirical even in axioms) is kept a dark secret, and children are often misled about it in school texts, which use plausibility arguments to motivate axioms.
The church itself was very sure that facts or observations (hence the real world) had to be avoided like the plague, especially after its adoption of reason or axiomatic proof. Thus, look at Aquinas’ “angel axiom”, that “angels occupy no space”. How will one check whether it is true or false in the real world? One cannot, because angels don’t exist in reality. One must accept the axiom on faith, and axiomatic reasoning will then tell us all about the resulting “relative truths” or faith-based fantasies that can be deduced.😀
According to Karl Popper’s demarcation, anything whose truth cannot be tested empirically is called metaphysics (=unreal), though “non-physics” would be a better word, to avoid the “polemic of superiority” implicit in the prefix “meta”. The church loved metaphysics (or the unreal); its reasoning about unreal angels just being the starting point. There were so many unreal things to reason about: God, heaven, hell, virgin birth. It kept medieval theologians busy for centuries!
Though Russell and Hilbert exposed the absence of axiomatic proofs in the “Euclid” book, they remained victims of the church polemic of the “superiority” of axiomatic reasoning, so “natural” for Westerners in those racist and colonial times. Therefore, paradoxically, Russell and Hilbert, while recognizing the absence of axiomatic proof in “Euclid”, converted all math to axiomatic metaphysics.[42]
Thus, the axioms of mathematics are equally metaphysics. For example, consider Hilbert’s axiom that there is unique straight-line through any two geometric points. Let us first think about a geometric point in the usual way as taught in school (as in the class VI NCERT text[43]) as an invisibly thin dot. If points have some finite size Hilbert’s axiom is obviously false: one can easily connect the two dots with more than one straight line. But saying that points have NO size makes points invisible AND metaphysical.
To clarify, the issue is not just the invisibility of geometric points. Viruses and bacteria too are invisible to the naked eye, but can be seen under a microscope. Electrons are even tinier, and cannot be viewed under a microscope, but a moving electron can be tracked in a bubble chamber, because an electron has some indirect physical effects. Therefore, invisible electrons are refutable on Popper’s criterion, hence physics. But invisible geometric points are metaphysics, for they have no refutable consequences.
A straight line is defined in school texts[44] (p. 71) as the infinite extension of a straight-line segment on either side. Though the text makes every attempt to deceive young children that a straight line (in formal math) is something plausibly physical it is forced to ask “Do you think you can draw a complete picture of a line? No. (Why?)” This is not just a question of visualization. Talk of infinitely long lines is metaphysics, and asserts unconcern for the real world. For, exactly how does one know that the real universe is infinitely large, so that infinitely long lines can really be drawn? In axiomatic mathematics, one can lay down any axiom one wants.
Of course, one could do geometry with other axioms, in an even more metaphysical way, with points and lines regarded as sets. But axiomatic set theory involves an even vaster metaphysics of infinity. For example, a line is represented by real numbers which are said to be uncountably infinite. Few, even among mathematicians, understand axiomatic set theory.[45] (We will not go into that here, since the immediate point is only that the axioms of mathematics are metaphysics. We will later touch upon exactly what kind of a metaphysics of eternity/infinity or whose metaphysics (or fantasies) they are.[46])
One can go on in this manner, pointing out the metaphysics in math. Thus, the notion of “distance” is commonly understood to involve physical measurement. (Measuring a curved line requires a string, not any instrument in a geometry box.) But if we use the notion of physical distance (and a straight-line segment is the shortest distance between two points) then there are no straight lines in the real world.
Theorems of axiomatic math are NOT true in the real world
Thus, lines drawn on the surface of the earth are necessarily curved, because the surface of the earth is curved. This means, the “Pythagorean theorem” fails for right angled triangles drawn on the surface of the earth. Such triangles were used in navigation up to the 18th century, and the European belief that the Pythagorean theorem is “true” let to numerous navigational disasters. In fact, space too is curved, so the theorem fails to apply exactly anywhere in the real world. In short, because the axioms of math are metaphysics, hence do not apply to the real world, the theorems of math (including the Pythagorean theorem) are only “relative truths” which do NOT apply exactly to the real world.
The typical dodge is to say the theorem applies “approximately” to the real world. But to speak of “approximation” is to admit the fallibility of the axiomatic method. Then the whole justification for infallibility fails. One should then abandon formal math and revert to normal math. Indians, for example, used normal math (ganita), and knew, long before “non-Euclidean” geometry, that it is “gross” (erroneous) to apply the “Pythagorean theorem” to navigation.[47]
However, the church (and axiomatic math) both dodge reality by making things metaphysical. For example, the notion of “distance” can be altogether avoided (as Hilbert did in his “synthetic” reformulation of “Euclidean” geometry), or it can be defined metaphysically. It is actually quite hard to metaphysically define the notion of length along a curved line: difficult to do if the line is jagged; usually defined for a “rectifiable arc”, using calculus.
Still, whatever way one defines distance, the notion of “shortest distance” involves a comparison with distance along infinitely many other lines: which comparison must be done metaphysically (in fantasy) for it cannot be done physically in finite time. So, again, all we have here is a metaphysics of infinity. And none of these various metaphysical reformulations, all used to axiomatically prove the Pythagorean theorem, will help to make the theorem true in the real world. (Those who argue in favour of metaphysics should note that the use of a biased metaphysics here is a con-trick intended to evade reality. This purpose of evading reality, where there is no need to, should be clearly stated.)
Unlike axiomatic geometry which is built around a pre-existing geometry (of normal math which accepts empirical facts, and applies to the real world) there are many cases in math, where the guidance of pre-existing normal math is NOT available. In these cases, axiomatic math goes wildly wrong (not just approximately wrong) in relation of its validly proved theorems to the real world.
For example, the Banach-Tarski theorem[48] says that a ball of gold can be divided into a finite number of parts which can be reassembled without stretching to obtain two balls of gold of the same size! That is axiomatic set theory leads to this nonsense get-rich quick scheme of converting one ball of gold into infinitely many balls just applying set theory. Of course, no one in the real world ever got rich this way. But that is “universal” mathematical “truth” for you. 😀
So, some mathematical theorems may totally fail to apply to the real world. Of course, formal mathematicians can easily invent excuses (non-Lebesgue measurability) for such failures, just as easily as astrologers can explain the failure of their prophecies. But we don’t need worthless metaphysical explanations of why the metaphysics of axiomatic math often goes so terribly wrong, and does NOT apply to the real world. The simple fact is that the purported rigor of “modern” math has reduced it to medieval theology.
Anyway, the bottom line is: (axiomatic) mathematical theorems typically are NOT valid knowledge in the real world. They are mere “relative truths” relative to metaphysical axioms, and a 2-valued logic which itself may fail to apply to the real world, as in quantum logic. There is no way to test whether the metaphysical axioms are true. The only way to decide “truth” in metaphysics is by authority.
People laugh at the medieval Christian theologians trying to fix the exact number of angels that could fit on a pin, but modern mathematics has turned a large number of university mathematicians into exploring a similar utterly useless metaphysics of infinity. In short, “modern math” has resurrected medieval Christian theology.
The aesthetic value of math
The above conclusion (that “theorems of axiomatic math are typically not valid knowledge in the real world”), and the earlier section on the lack of practical value of axiomatic math, raises a serious question-mark about axiomatic math. Why do it (useless axiomatic math, NOT useful normal math)? Why prove theorems axiomatically?
Hardy[49] admitted, “mathematics is… unprofitable. Is mathematics… useful, as other sciences such as chemistry…are? …I shall ultimately say No”. Why, then, do (axiomatic) math? Hardy’s answer: “A mathematician, like a painter or a poet, is a maker of patterns…The mathematician’s patterns…must be beautiful; Beauty is the first test” (pp 13–14) (This is dishonest: when it comes to funding, mathematicians don’t really seek it from the department of culture.[50])
Further, is “beauty” too defined axiomatically? Hardy continues, by appealing to intuition, “It may be very hard to define mathematical beauty, but…that does not prevent us from recognizing… it.” Such appeals to “intuition” are a warning sign of impending con-tricks of metaphysics.
Thus, Hardy is guilty of an inexcusable error: he confounded two distinct notions of geometry: (1) Egyptian mystery geometry, as articulated through mathesis by Plato and Neoplatonists, with (2) the church-origin formal math which is axiomatic metaphysics. This confusion is common among chauvinistic Westerners (including Russell) who gullibly swallowed the Crusading and racist history of the Greek origins of axiomatic math, which we have seen (in the first part of this article) is false. Mathematicians, like Magister Ludi, may enjoy the glass bead game of the sterile metaphysics of Bourbaki, and declare it beautiful, but children know better that the emperor has no clothes.
Children intuitively do understand what Hardy didn’t: the difference between the two types of math, as also the ugliness (and uselessness) of church-origin formal math. Hence, they reject it in such large numbers today. Plato, in his Republic,[51] prescribed both math and music, as essential to the education of the young men in the Republic, because both arouse the soul and Plato believed that makes people virtuous. And the same children today, who reject (ugly, axiomatic) math, all love the beauty of music.
The rejection of present-day (axiomatic) math by children in large numbers is an observed fact, which proves its ugliness, unlike Hardy’s “intuition”, tied to his means of livelihood. But children in large numbers still enjoy music. That too is an observed fact. What was the change since Plato?
The change that took place in Western math, is fatal to the Western glorified narrative of Greek-origin of mathematics, hence never mentioned by any Westerner. The church, in its first religious war with “pagans”, cursed Plato’s notion of soul (because that notion supported equity). As such, church-origin axiomatic mathematics was fundamentally different. Its primary political aim was that of the church: persuasion for conversion to fill its coffers, NOT soul arousal. Why should any ordinary human being be concerned with the related metaphysics used to browbeat the listener? Hence, children find formal math ugly.
Formal mathematicians waste their life proving theorems, which are neither useful nor beautiful. Then, they talk of the beauty of (formal) mathematics as yet another easy way to con people, as in the story of the emperor’s new clothes.
Summary and conclusions
The church intervention in math co-originated with the myth of “Euclid”: but there are no axiomatic proofs in the actual “Euclid” book, whoever wrote it. That was just a motivated misreading of the book.
How, then, did axiomatic proofs actually originate? Who FIRST advocated axiomatic proofs? The Crusading church did. Axiomatic proofs originated during the Crusades (shortly after the “Euclid” book arrived in Europe as a crusading trophy). Why?
Because the church had a political requirement for reason, during the Crusades. But traditional normal reasoning, starting from facts, threatened church dogmas. So, while accepting reason, the church subtly twisted it to turn reason into a church tool. How? By excluding facts (which threatened its dogmas) and thus inventing axiomatic reasoning. The novel aspect of axiomatic reasoning is NOT the mere use of reasoning, which is ancient, but the prohibition of the empirical (facts, observations).
Its advantage? Church dogmas contrary to facts are safe. But that is an advantage only for the church. Further, by selecting axioms/assumptions appropriately, one can reason and prove ANY pre-desired conclusion whatsoever. This is equally true in axiomatic math today.
It is asserted that axiomatic proof is “superior”. In what way is axiomatic math superior? Axiomatic proof does not add superior practical value, which all still comes from normal math, whether in a grocery shop or to calculate rocket trajectories or for machine learning.
Axiomatic proofs are epistemically inferior: deductive proof is more frequently fallible. Also, even when valid, axiomatic proofs lead to mere “relative truths” (relative to axioms and logic). That is (axiomatic) mathematical theorems typically are NOT valid knowledge in the real world.
There is no way to judge the truth of axioms (hence theorems), except by authority, for axiom of math are mostly a (non-empirical) metaphysics of infinity. Consequently, some theorems of mathematics may be wildly untrue in the real world.
Children’s rejection of formal math shows it is ugly. Formal mathematicians spend their life proving theorems, which are neither useful nor beautiful (nor even secular, since the underlying metaphysics is biased, as we will see).
References
[1] “Racism in the math classroom: “Pythagorean theorem” and the two myths of “Euclid”.
[2] B. Russell, ‘The Teaching of Euclid’, The Mathematical Gazette 2, no. 33 (1902): 165–67.
[3] David Hilbert, The Foundations of Geometry (The Open Court Publishing Co., La Salle, 1950).
[4] Thomas Aquinas, Summa Theologica, First Part, Q. 52, article 3.
[5] 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).
[6] A key European Muslim exponent was Averroes (Ibn Rushd). As is well known, his books were widely used as textbooks in the first European universities. Some further details: C. K. Raju, ‘How to break the hegemony perpetuated by the university: decolonised courses in mathematics and the history and philosophy of science (Arabic)’, in Culturalisation of Humanities: Vision and Experiments. (Proceedings of the International Conference on Culturalization of the Humanities, Beirut, November 2018.) (Beirut: Al Maaref University, 2019), 77–114, Also, (English) Studies in Humanities and Social Sciences 26, no. 2 (2019): 86–109.
[7] C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003) chp. 4: ‘Newton’s secret’.
[8] C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant of Racist and Civilizational Superiority — Part 1’, Arụmarụka: Journal of Conversational Thinking 1, no. 1 (2021): 127–55.
[9] C. K. Raju, Is Science Western in Origin?, Dissenting Knowledges Pamphlet Series (Multiversity, 2009).
[10] The use of reason as a means of proof is documented in the Indian Nyaya sutra (verse 2). It was used in traditional Indian math. For a more detailed explanation, see this video. The use of reason for proofs pre-dates the Buddha, since the Lokayata rejected it. Satish Chandra Vidyabhushana, The Nyaya Sutras of Gotama (Allahabad: Pāninī Office, 1913).
[11] conflated with the real Aristotle of Stagira, C. K. Raju, ‘Logic’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016 2008).
[12] E.g., NCERT class IX school text, Appendix 1: “(We do not distinguish between axioms and postulates these days) [p. 296] An axiom is a mathematical statement which is assumed to be true without proof. [emphasis original, pp 297–98]”. .
[13] Josiah Priest, Bible Defence of Slavery: To Which Is Added a Faithful Exposition of That System of Pseudo Philanthropy, Or Fanaticism, Modern Abolitionism … and Proposing a Plan of National Colonization (W.S. Brown, 1851).
[14] The Cambridge “Newton” scholar D. T. Whiteside was annoyed by my exposures of Newton’s critique of the church. The fact is that Cambridge historians kept this fact hidden for two centuries (though Newton’s religious beliefs influenced his science). Therefore, Whiteside responded by lying about Newton’s work, and abusing me on the mailing list Historia Matematica in 1999, because abuse is the perfect way to demonstrate “superiority”! For my exposure of Newton, by now well known, see Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs chp. 4, Newton’s secret.
[15] Oxford university was alarmed by my critique of the formalist philosophy of math, and appointed a LITERALLY illiterate “reviewer” who not read beyond chap. 2 of my book, since that requires familiarity with at least Sanskrit. Of course, my book is on the history and philosophy of calculus, in which the reviewer is an utter ignoramus, and of course he no understanding of the Indian philosophy of math. This illiterate reviewer was deemed the right person to review my, since the editor’s simple objective of soliciting the book for review was to personally attack me and thus try to discredit me and my book. Obviously, the standard of reviewers appointed in secret must be even worse! Oxford scholarship is still stuck in the days of the Inquisition. See, e.g., “Oxford must fall”.
[16] The Harvard professor Witzel was alarmed by my paper on the origin of probability in ancient India. To try to put me down, he told several brazen lies about me and my article (e.g., that I erroneously spoke of cubes or 6 faced-dice in ancient), though it could be easily checked that there is no mention of cubes or six face dice in my article. (I wrote of the 5-faced bibhitaka fruit used as dice.) He added that I used Wilson’s translation of the Rgveda. This was another lie which could again be cross checked and proved false in a few seconds. That Witzel could tell such numerous and brazen lies, shows that he has long practice in telling silly lies (about Sanskrit and India), and was confident he would get away unscathed. In support of Witzel’s lies, the editor tried to censor me. Indeed, he did not publish my response until I threatened to denounce him publicly at a conference. This again shows the kind of lumpenism prevalent at the highest level of Western “scholarship”. http://ckraju.net/blog/?p=56.
[17] In this case, the lumpen “scholar” was G. F. R. Ellis, of Cape Town, co-author of Stephen Hawking’s serious book, and Templeton prize winner, for showing how science can be used for the benefit of the church. Well aware of my critique of Hawking’s mathematical inadequacies, Ellis trembled at the thought of publicly debating even my published critique of Stephen Hawking’s creationism, in his own math department. See the section of 3.2 on “Stephen Hawking and singularities”, in my prior written summary, which technical part I intended to discuss in more detail in the math department, in the University of Cape Town. The formal math involved was technical, and far beyond the comprehension of Ellis, who knew I had debated it twenty years earlier, in 1997 with the mathematician Roger Penrose. Therefore, as befits a church propagandist, Ellis launched a slimy tirade against me to stop me from speaking in his math department and exposing him. He “cleverly” used his Indian student to plant all sorts of abuses and lies against me in the racist South African Press. See, Adam Cooper, “Surely good scholarship means having our perspectives challenged”, Daily Maverick, South Africa, 11 October 2017. Note how some of the slime was accepted as true even by Cooper, a sympathizer, who thought I, not Ellis, was the one who did not know math! For the latest on Penrose and Hawking’s creationism, see C. K. Raju, ‘A Singular Nobel?’, Mainstream 59, no. 7 (30 January 2021), http://www.mainstreamweekly.net/article10406.html.
[18] C. K. Raju, ‘To Decolonise Maths, Stand up to Its False History and Bad Philosophy’, The Wire, 2016. C. K. Raju, Mathematics, Decolonisation and Censorship, Kafila, 2017, ; C. K. Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’, Journal of Black Studies 48, no. 3 (2017): 256–78, https://doi.org/10.1177%2F0021934716688311.
[19] For an analysis of why such “arguments” are (non-verbally) persuasive, see Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs, Appendix: ‘Patterns of irrationality’.
[20] Think of this assertion of Western inferiority as “affirmative action” or “reverse discrimination”. This is badly needed to balance the centuries of repeated assertion of White/Western intellectual superiority. Those claims were earlier based on intellectual theft using the doctrine of Christian discovery, etc. That intellectual theft continues today, as in the serial plagiarism of my work by the top mathematician, Michael Atiyah. And “leading” Western institutions, like the American Mathematical Society (AMS) not only cover it up, they invest in propagandists, who then help to bury the unethical cover-up by the AMS of that serial plagiarism.
[21] Plato, Meno, trans. Benjamin Jowett, (Search for the third occurrence of “soul”.)
[22] Plato, Phaedo, trans. Benjamin Jowett, . (Search for “diagram”.)
[23] Proclus, A Commentary on the First Book of Euclid’s Elements, trans. Glenn R. Morrow (Princeton, New Jersey: Princeton University Press, 1970).
[24] Russell, ‘The Teaching of Euclid’.
[25] Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars; Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs chp. 2, “The curse on ‘cyclic’ time’.
[26] “How to make calculus easy”.
[27] Note that the real number 1 is NOT the same entity as the natural number 1 and Peano’s axioms cannot be applied to prove 1+1=2 in real numbers as the mathematician in the Cape town debate on decolonisation of science wrongly thought. The axiomatic proof of 1+1=2 in real numbers is a lot more complicated, involving axiomatic set theory etc.
[28] For a simplified account, see C. K. Raju, ‘California, Indian Calculus and the Technology Race. 2: Don’t Cancel the Calculus, Make It Easy!’, Boloji.Com, 24 December 2021; for an actual computer program 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): 325–62. For a fuller account of floating-point numbers, see extract from my old lecture notes on computer programming in C.
[29] C. K. Raju, Statistics for Social Science and Humanities: Should We Teach It Using Normal Math or Formal Math?, JNU, Sep 2020.
[30] C. K. Raju, ‘Decolonising Mathematics’, AlterNation 25, no. 2 (2018): 12–43b.
[31] Raju.
[32] https://twitter.com/CKRaju14/status/1494113441451474944.
[33] C. K. Raju, ‘Junction Conditions in General Relativity’, Journal of Physics A: Mathematical and General 15 (1982): 1785–97.
[34] Nobody has, to date, empirically tested the final resulting conditions for relativistic (and non-relativistic) shock waves in real fluids, such as air or water which has viscosity and thermal conductivity. C. K. Raju, ‘Distributional Matter Tensors in Relativity’, in Proceedings of the 5th Marcel Grossman Meeting, ed. D. Blair and M. J. Buckingham (World Scientific, 1989), 421–23, arXiv:0804.1998.
[35] G.H. Hardy, A Mathematician’s Apology (University of Alberta Mathematical Sciences Society, 1940), 11.
[36] https://www.youtube.com/watch?v=d85UOuY0DFU.
[37] Haribhadra Suri, ed., षटदर्शन समुच्चय, 5th ed. (Bharatiya Jnanapeeth, 2000) famous e.g. of Wolf’s footprints, karika verse 81, commentary 559, page 452.
[38] Bertrand Russell, ‘Mathematics and the Metaphysicians’, in Mysticism and Logic and Other Essays (London: Longmans, Green, and Co., 1918), 74–96.
[39] Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’.
[40] http://ncert.nic.in/textbook.php?iemh1=a1-15.
[41] Raju, ‘Decolonising Mathematics’.
[42] Russell, ‘Mathematics and the Metaphysicians’.
[43] http://ncert.nic.in/textbook.php?femh1=4-14, chp. 4, p. 69–70.
[44] “Imagine that the line segment from A to B is extended beyond A in one direction and beyond B in the other direction without any end (see figure). You now get a model for a line.” http://ncert.nic.in/textbook.php?femh1=4-14,
[45] For a critique of the axioms of set theory, see http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-2.pdf.
[46] 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.
[47] Specifically, Bhaskar I (7th c.) says (Mahabhaskariya 2.5), the method of plane navigation (effectively using the “Pythagorean theorem”) is gross. http://ckraju.net/papers/presentations/images/Maha-Bhaskariya-2-5-trans.pdf. Bhaskar, Bhaskar I and His Works, Part 2: Mahabhaskariya, ed. K. S. Shukla (Lucknow: Dept of Math and Astronomy, Lucknow University, 1960).
[48] http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-2.pdf.
[49] Hardy, A Mathematician’s Apology, 8.
[50] C. K. Raju, ‘Kosambi the Mathematician’, Economic and Political Weekly 44, no. 20 (16 May 2009): 33–45.
[51] http://classics.mit.edu/Plato/republic.8.vii.html, search for “very little of either geometry”.
