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Mainstream, VOL LIX No 7, New Delhi, January 30, 2021

A singular Nobel? | C. K. Raju

Friday 29 January 2021, by C. K. Raju

Introduction

On the common story, science is at war with all religions. This neglects the fact that present-day (Western) science developed under the hegemonic rule of the church. Consequently, present-day (Western) science is influenced by church theology. For example, we still speak of Newton’s “laws” of motion on the belief that the world is governed by eternal laws of nature. This belief is NOT scientific on Popper’s criterion of refutability or falsifiability. In fact, it is anti-scientific, for the belief in “laws” militates against the human creativity needed to design novel experiments to test scientific theories. [1] But this belief, that God rules the world, using eternal laws of nature, was a Christian theological belief, first articulated by Aquinas. [2] It is contrary to certain key Islamic beliefs such as those of al Ghazali [3] that Allah has habits which may change. This is often taken to mean not that the church has influenced science, but that Christianity is more scientific than Islam! [4]

Stated in non-theological terminology, al Ghazali’s point is that all we observe are certain regularities which may keep changing with time. This resonates with the Buddhist observation that nothing in the world endures for even two instants. The fact of the matter is that most people who naively believe this story of a war between science and religion are unfamiliar with even elementary science, and know little of the theology or philosophy of various religions.

Though colonial education ostensibly came for science, the fact is that the vast majority of the colonized elite are mathematically and scientifically illiterate. This fact about the scientific illiteracy of the colonially educated was brought out very clearly in a recent lecture at JNU, [5] when I posed my "Cape Town" challenge to the JNU faculty. The challenge is to prove 1 + 1 = 2, in formal real numbers. (“Real” numbers are taught in class IX, and in all our universities as essential for calculus, which is needed to write down the equations of physics.) This challenge was backed by a reward of ₹10 lakhs for anyone who could meet the challenge in one day, and a reduced reward of ₹ 1 lakh for anyone who met in a week, though nobody did. This shows the level of scientific illiteracy among the faculty in our most prestigious universities: the situation in Delhi University or IIT’s is not very different.

Under such conditions of extreme ignorance, created by colonial education, uncritical acceptance of scientific authority has come to be regarded as "scientific temper". Likewise, any critique of scientific authority is treated as “unscientific” and censored, by the ignorant, both within science and by the journalistic world. This is not scientific temper this is scientific idolatry.

This blind belief in scientific authority obviously allows it to be misused to manipulate popular beliefs. A key case in the point is the creationism of Stephen Hawking. Once again, the popularity of Stephen Hawking rests on the underlying claim that "science has proved the truth of Christianity", as so explicitly and lengthily stated by Tipler etc. I have written extensively on the contrary position that church politics has influenced science, in my book The Eleven Pictures of Time, [6] and related popular level articles [7] etc.

Hawking’s creationist claims involved singularity theory initiated by Roger Penrose who received the Nobel Prize in physics last year, for singularity theory. It is noteworthy that during my two-day debate with Roger Penrose, way back in 1997, though many scientists from Delhi University and JNU were present, not a single one ever publicly took a stand against even Tipler, leave alone singularity theory. For the colonised, the thought of Western superstitions is taboo, for it is an act of disloyalty, a revolt against the intellectual authority of the master and his scientific icons. Apart from me and Penrose, no one else present understood all the formal mathematics involved.

These uncomfortable aspects of singularity theory are what the present article proposes to discuss in "long form". The long form is needed since the attempt is to try to make the issues clear even to those (almost all people) who lack a technical background, since (unlike blind acceptance of praise) a critical view is not possible without understanding. This is a novel attempt, because in India, there is no earlier example of "popular science" which is critical of the establishment.

Such a critical tradition needs to be fostered because science is not about faith, scientific authorities may be untrustworthy. Thus, a few years earlier, in 2017, I attempted to debate singularity theory with Stephen Hawking’s co-author G. F. R. Ellis, during a visit [8] to the University of Cape Town. These attempts to debate were met not by a single scientific counter-argument, but censorship, in the manner of the worldwide censorship of my earlier article on decolonizing mathematics. [9] This refusal to debate was justified by scurrilous attacks against me in the racist South African press, promoted by Ellis and his supporters. Those attempts to demonise me have been transported through church journalists to North America. [10] That is, not only does church theology creep into science, the resulting untenable conclusions are defended as “scientific” by the same old church methods of censorship and demonization of all critics. As the church demonstrated these methods can be used to persistently defend any number of nonsense claims such as virgin birth. Reasoned debate is the one thing that is prohibited.

This creep of theology into Western science is facilitated by formal mathematics, or the mathematics that colonial education brought, [11] which is laced with church superstitions and myths, but passed off as “superior”. Formal mathematics is divorced from the empirical: [12] this divorce does not suit science but it greatly suited the church method of reasoning without facts, as in church rational theology. For example, Aquinas deduced that many angels may sit on the head of a pin; since there are no facts about angels this deduction began from the a postulate that angels occupy no space. As the Lokayata pointed out thousands of years before Aquinas, any nonsense can be proved in this deductive manner.

To the contrary, Western thought promotes the superstition that such deduction, based on reasoning without facts, is infallible. [13] In the 20th century, when it was finally accepted, after eight centuries of gullibility, that it is a myth that there are axiomatic proofs in Euclid’s Elements, David Hilbert and Bertrand Russell nevertheless based formal mathematics on this superstition about the infallibility of deduction (minus facts, and based on metaphysical axioms alone). Understandably, it is very hard for the colonized to bear this fall of their icons, especially when all they have learnt is scientific idolatry.

But it would be great if rationalists (and all those offended by my critique) were to respond rationally—after due study—instead of using the church techniques of censorship and demonization, as the sole possible defence against this critique of churchified rationality. It would also be great if secularists were to publicly oppose (or defend) the intrusion of church myths and superstitions in a compulsory subject like school mathematics, after colonial education. [14] That intrusion also glorifies the coloniser but only ends up making mathematics impossibly difficult (remember 1+1=2) without adding an iota of practical value for science. (That this way of teaching mathematics adds political value for the colonizer, against the colonized, is another matter.)

Contents

Part 1: The theology and politics of singularities
Hawking was an atheist
The deep parallels between Augustine and Hawking
Is a singularity a robust consequence of general relativity?
Part 2: The formal math and empirical consequences of singularities
The formal mathematics of singularities
Failure of physics or failure of math?
Conservation laws
Riemann’s mistake
Products with the Dirac delta
Ignorance of math among singularity theorists
Part 3: Singularity theory and the decolonisation of math
The challenge to formal mathematics
Church myths and superstitions in formal mathematics
The consequences of colonial education
Censorship

Part 1: The theology and politics of singularities

That science, in practice, is not so much about testing, as it is about trusting the authority of scientists and their sponsors, has been made amply clear by the latest rollout of an untested vaccine. The Nobel prize is the ultimate in scientific authority. But, that even the Nobel prize is mixed up with politics is nothing new. Most people understand that with regard to the Peace prize and the Economics prize. But people believe that Nobel prize for the hard sciences is free from politics.

Of course, it is not free from individual-level politics. The late E. C. G. Sudarshan used to complain, every time we met, that he was twice robbed of a Nobel prize. He used to take solace in my argument that Einstein stole credit for special relativity from Poincaré, and the credit for general relativity from Hilbert. The common tendency, as in the case of Laplace, was for famous persons to rob credit from the not-so famous. But, the novelty of Einstein’s case was that both Poincaré and Hilbert were already famous mathematicians of their time, compared to Einstein who was then unknown. Both understood that Einstein was an ignoramus, quite contrary to his present-day popular image. Hilbert even explicitly declared “Every boy in the streets of Goettingen knows more about four-dimensional geometry than Einstein.” [15] However, Hilbert was over-smart and attributed the striking originality of the theory of relativity to Einstein’s ignorance (and not Poincaré’s brilliance). Poincaré was nominated for the Nobel prize by numerous people including Marie Curie, but was denied the prize on the ground that it is not given for mathematics. Possibly for the same reason, Einstein was not given the Nobel prize for either special or general relativity which latter was considered a part of mathematics, and used to be taught mostly in university departments of mathematics in India until the 1980’s.

At any rate, people believe that the Nobel prize in science is not given to promote the same kind of institutionalised politics brazenly promoted by the Peace prize or the Economics prize. Why not? Not because the Nobel prize committee is incapable of politics, but because people believe there is no scope for politics in science beyond credits and attributions, not, at least, in the substance of science as there is in economics.

But last year, Roger Penrose got half of the Nobel prize in physics. The citation says it is for “the discovery that black hole formation is a robust prediction of general relativity”. However, there is undoubtedly some subjectivity here, for what Penrose actually proved was only that, on general relativity, gravitational collapse must give rise to a singularity. Is a singularity necessarily a black hole? The question about the physical interpretation of a singularity needs to be asked, since there is no empirical proof of a singularity.

In my two-day debate with Roger Penrose, in Delhi’s India International Centre in 1997, [16] I first pointed to the allied but striking theological interpretations of a singularity.

Thus, Stephen Hawking applied singularity theory to the whole cosmos to conclude that a cosmic singularity proved that the big bang involved one-time creation, an initial moment where the “laws” of physics break down. (The big bang is not, in itself, necessarily a beginning of time or a moment of creation, for it could have been preceded by a “big crunch”.) A singularity is necessary to speak of creation. This is the concluding sentence of Hawking’s SERIOUS book on general relativity [17] (p. 364).
“the actual point of creation, the singularity, is outside the scope of presently known laws of physics” (emphasis added).

The resemblance of a singularity to the Christian God is not incidental. A science journalist persistently asked Hawking: what happens at a singularity? The persistent answer was “since the laws of physics break down, anything at all could happen.” Despite repeated questioning, Hawking stuck to that answer. So, the science journalist explained it thus to his audience: what would one do if one came face to face with a singularity? Since anything at all could happen, what was there to do but to go down on one’s knees and pray!

In his popular-level writings, Hawking clarified that the breakdown of the “laws of physics” had an important theological meaning: it meant that (the Christian) “God would have had a free hand to create the world [and would not be constrained by the “laws” of nature]” His co-author G. F. R. Ellis, from apartheid South Africa, even got the million-dollar Templeton award for thus using singularity theory to claim that science had proved the truth of the Christian (or rather the church) view of creation.

F. J. Tipler, who boasts of having published many articles in the authoritative journal Nature, went a step further. By 1994, over two years before my debate with Penrose, Tipler had already published a book [18] (published by the Macmillan, the publishers of Nature) explicitly claiming that singularity theory had proved the truth of all Judeo-Christian theology as physics. This may sound incredible, especially to the large number of those colonised who believe the excessively naive story that science has always been at war against the church since Galileo. Accordingly, here is a direct quote from Tipler asserting that his is a book purporting to show that the central claims of Judeo-Christian theology are in fact true, that these claims are straightforward deductions of the laws of physics as we now understand them. I have been forced into these conclusions by the inexorable logic of my own special branch of physics...the area of global general relativity...created...by the great British physicists Roger Penrose and Stephen Hawking.

What a great victory for science in its supposed war with religion! The indoctrinated colonised cannot dream of the opposite possibility that the church co-opted science (in the West), as Roger Bacon explicitly suggested it should, to encourage the spread of Christianity.

When I brought up these theological implications of singularity theory during the 1997 debate, Penrose responded evasively to my critique. He did NOT take a straightforward stand and did NOT openly condemn Tipler. (He perhaps thought no one would notice this evasiveness; but we, the colonised, have had numerous experiences of British duplicity.) Instead, Penrose urged me not to talk of the interpretation of a singularity, but to focus only on his mathematical proof. Why is the mathematical proof of the slightest importance if not for its physical interpretation? And the Nobel prize has been given to Penrose solely for his interpretation of a singularity as a black hole, assumed to be correct by the committee, and not for its mathematical proof, since the Nobel prize is not given for mathematics.

So, after the award of a Nobel will all the Christian theological beliefs associated with a singularity just go away? Or, to the contrary, will that prize embolden creationists and smarter intelligent-design theorists like Ben Carson? Was this perhaps the hidden political purpose behind the award of the prize to such a doubtful case? Given the politics accompanying the other prizes, we can hardly rule out this possibility. People may not believe in Christian theology, but they implicitly believe in science, especially if it is endorsed by the Nobel prize, and especially if they are ignorant about it.

Hawking was an atheist

Many people regard Hawking’s defence of Christian creationism as credible for a peculiar reason unconnected with science: they say he was an atheist. While this is admittedly a great propagandist argument which misleads numerous people, it is actually unacceptable. Both Jinnah and Savarkar were confirmed atheists. Does that make all the religiosity and killings associated with their politics and the partition just disappear? (Of course, I am not absolving the British, and agree that the partition involved a religious quarrel which the British precipitated and instigated with all their might, on their policy of divide and rule.)

But the immediate point is this: are people willing to say that the Hindu Mahasabha or the Muslim League had no connection to religion just because the founders were atheists? And are they willing to accept the possible religious bias in any science done by church believers like Ellis? If not, they should not apply a different standard in the case of Stephen Hawking. Instead, they should simply accept that Hawking’s personal atheism or otherwise is completely irrelevant to his opportunistic scientific and political support for Christian creationism. More so, because the message received by millions, who bought his popular-level book, was a message drenched in Christian religiosity. (Of course, the colonised will never give up their story of science and religion at war, and will keep citing the one story of Galileo, from 500 years ago to prove their case, without ever having studied the facts of even that one case.)

Anyway, it should be very clear to anyone who has studied church history (though few colonised, none that I know, have written about it) that from the 4th c. onwards, the church was not so much about religion or belief in God, as it was a political organization to attain and retain political power through superstitions. Belief in God was just one tool for that method of power through superstitions: uncritical belief in “science” so long as it supports the church, works equally well to that end of power.

The deep parallels between Augustine and Hawking

As regards Hawking, I should also clarify that the theological association of singularity theory are not merely a matter of the expository use of popular metaphor. The theological parallels go much deeper. My book Eleven Pictures of Time (Sage, 2003) explained, for the layperson, how Hawking’s entire argument about time from initial assumptions, to creationist and apocalyptic conclusions about a singularity, has striking parallels to the Christian theologian Augustine’s arguments about time. As my book further explains, those time beliefs are at the foundation of (politicised) post-Nicene church theology—accepted by all Western Christian denominations, including Protestants, of course. These are the beliefs at the heart of the notion of the “West”, as elaborated by Toynbee, Samuel Huntington etc., who first championed those cultural beliefs as a key part of Western politics, and they are a key part of Trump’s stated agenda.19 [19]

For example, Hawking and Ellis needed a key assumption to “mathematically prove” the existence of a cosmological singularity. They call that assumption the “chronology protection postulate/conjecture”. Briefly, it says that any kind of cyclicity in time is prohibited. That is exactly what Augustine did: he proscribed any kind of “cyclic” time by fiat. The church fully endorsed Augustine, by recognizing this assumption as critical to its post-Nicene theology, and by pronouncing its great curse (anathema) on cyclic time during the fifth “ecumenical” council held at the historic Hagia Sophia in 552 CE. [20] The key point: Hawking’s proof of a singularity is not based on just general relativity, or “science” but involves an assumption about the nature of time which is essential to church politics. Without such an assumption, there would be no Hawking singularities. And the Lokayata have long warned us that bad assumptions lead to false conclusions.

And how was that assumption justified? What was Augustine’s argument in favour of his fiat? As a first step, as a true “saint”, he misrepresented and deliberately confounded two distinct types of cyclic time: eternal recurrence and quasi-cyclic time were both conflated into a single category of ‘cyclic’ time. Augustine surely hoped to confuse ordinary people by this misrepresentation, for this distinction between different notions of cyclic time was well known in his time, and explicitly emphasized by the earlier theologian Origen to whom we owe the current Bible. I have since long put up quotations from Origen’s De Principis, [21] on my website, since this is a confusion in Western thought about “cyclic” time, a confusion which recurs eternally.

Then, Augustine argued (using this motivated misrepresentation of Origen) that the endless repetition of Christ’s death would imply a destruction of the Christian notion of morality. On that notion of morality, Christ’s suffering was meant to save the world. But with eternal recurrence, Christ would repeatedly suffer, and yet would be unable to save himself, leave alone the world, from suffering. To reiterate, this was a motivated misrepresentation, since Origen had explicitly stated in his De Principis (see above quotations) that Christ did not suffer in previous lives. (Note that immediate discussion concerns the politics of rebirth or “cyclic time” in Christianity, not its science.)
The real reason for Augustine’s saintly lies, as I have explained, was a political fear of equity: the then major opponents of the Christians, the “pagans”, believed in equity and that all would eventually be saved. Origen championed this belief, and saw “cyclic” time as a source of equity. But the post-Nicene church wanted to peddle the fundamental superstition that Christians would get preferential treatment in the after life. Jerome, who was the author of the Latin Bible (the Vulgate), based on Origen’s notes, stated this explicitly

Now I find among the bad things written by Origen the following: that there are innumerable worlds, succeeding one another in eternal ages...that in restitution...Archangels and Angels, the devil, the demons and the souls of men whether Christian, Jews or Heathen will all be of one condition and degree [i.e., they too will be saved], and...we who are now men may afterwards be born women, and one who is now a virgin may chance then to be a prostitute.

It was this firm belief in inequity (or the superiority of Christians) which led to the great crimes against humanity by the church, such as genocide on three continents, or racism, crimes which are celebrated today instead of being condemned.

Hawking imitates Augustine’s fiat: because a mathematical axiom or postulate is just that, a fiat. He introduces the chronology protection postulate that there ought be no cyclicity in time. To support this postulate, Hawking fully imitates Augustine’s dishonest method of misrepresentation, by confounding any kind of cyclic time with eternal recurrence. His argument differs from Augustine in a minor detail: instead of speaking of the destruction of Christian morality, he talks of the destruction of science, but on similar wrong grounds that any kind of recurrence (since confounded with eternal recurrence) would destroy the “free will” needed to test science using experiments.
As I have explained, exactly contrary to the Augustine-Hawking belief that cyclicity is contrary to “free will”, a tilt in the arrow of time (which implies time travel and possible cyclicity) is actually the way to establish spontaneity (“free will”) in physics. [22]

Is a singularity a robust consequence of general relativity?

Two key points need to be reiterated. First, a singularity is NOT a robust consequence of general relativity alone, as stated in the citation of the Nobel prize committee. Certain deeply prejudiced assumptions about time have been slipped in, though few have noticed them. That is, Hawking’s singularities are a consequence of general relativity PLUS certain assumptions about time, assumptions strongly supported by the church to promote its iniquitous politics. These assumptions are religiously biased for they are against the beliefs about time in other religions (including early Christianity). But the ignorant don’t see those assumptions tucked away in a long book, and fail to understand how they are used to promote superstitions, the source of church power. The belief that singularities are a consequence of general relativity alone is one such superstition.

Secondly, Hawking’s argument (to support the extra assumption about time which he needs to prove the existence of a singularity) is exceptionally foolish. Thus, the argument cites the “free will” needed to perform scientific experiments. Yes. That mundane creativity or “free will” is undoubtedly a part of mundane experience. But where in the equations of general relativity is there a factor including that mundane experience and indicating “free will” of humans or other living beings? There cannot be: for human beings or life are not even defined in general relativity. So, if one accepts general relativity, as a “universal and eternal law of nature”, one must reject “free will”, for God’s laws or the equations of physics brook no exceptions, not at least as they are currently formulated. [23]

That is, Hawking, like theologians, deliberately uses contradictory assumptions: he accepts general relativity, but talks of “free will” when convenient, so as to justify certain assumptions, involving religious prejudices about time—assumptions which he needs to “mathematically prove” certain pre-desired conclusions. (And, obviously, the conclusion of a singularity relates to Augustine’s idea of creation and apocalypse, and the curse on cyclic time, as noted by Tipler, in an article [24] published in the journal Nature.)
And it is a well known principle of two-valued logic (which is used in the current method of mathematical proof) that any desired conclusion, whatsoever, may be derived from contradictory assumptions, which is why theologians use them so often.

In short, the plot of the narrative of a singularity doesn’t hold together any more than that of a bad Hollywood or Bollywood film. (It is another matter that physics can be easily corrected to allow for spontaneity, as I have shown, by using functional differential equations with mixed-type deviating arguments.) Science, especially cosmology, is ultimately about Western social acceptance, not debate or reliance on experiments or even the mundane experience that we continuously create a bit of the future. Continuous creation may be part of Buddhism or Islam but is no part of Christianity. (John Duns’ ideas in this regard, in Christian theology, were politically defeated by the "rationalists" led by Aquinas.) And the Nobel prize is the ultimate symbol of that Western social acceptance.

The theological ingredients and political implications of singularity theory are clear enough to be a troublesome matter (and an opportunity for those who want to promote a particular religion). But is singularity theory any sort of science? This was the other question I raised in my 1997 debate with Penrose.

Part 2: The formal math and empirical consequences of singularities

Setting aside ALL its theological ramifications, explained in the first part of this article, we can examine singularity theory from an empirical perspective.

Is the claim “a singularity is a black hole” any sort of science? Thus, on Popper’s criterion of falsifiability or refutability, science should have some (empirically) refutable consequences. What, if any, are those? Obviously, there is no empirical proof of the existence of a cosmological singularity, or a beginning of time (as distinct from a big bang or a mere compact initial state of the cosmos). Stephen Hawking, on his visit to Delhi, lamented that he had not got the Nobel prize. He did not get it for this very reason, that a cosmological singularity has no clear empirical consequences, different from those of a big bang. Likewise, the observation of a massive compact object is no proof of a black hole and its peculiar feature of an event horizon, for massive compact objects could easily also exist even in Newtonian gravitation as distinct from general relativity, [25] but the event horizon is what is peculiar to relativity.

Now, the Nobel prize citation imagines Penrose’s singularity as something at the centre of a black hole. (That is, surrounded by its so-called event horizon.) But, observing gravitational collapse from the viewpoint of an outside observer, far away (asymptotic observer), a black hole (or, more precisely, the event horizon) would take an infinite amount of time to form. (A black hole has other magical properties: for example, though its surface area is finite, its volume is infinite.) But an outside observer would be unaware of all this, hence unable to check any of this. He would have no empirical way to observe its infinite volume, for example. At best what an outside observer would “see” would be a tiny amount of light which might escape from a highly red-shifted surface. Such highly red-shifted surfaces have not been observed. The most highly red-shifted objects known, quasars, are believed to be red-shifted on account of the expansion of the cosmos (Hubble red shift), since far away (although that belief may be false).

Anyway, what we see (or don’t see) from outside a black hole would offer no clue as to what happens inside the event horizon, whether, for example, quantum gravity takes over, and prevents the formation of a singularity, as many believe. What we observe from outside would be just the same whether or not a singularity ever forms inside. Therefore, Penrose’s singularities are not empirically refutable.

Indeed, Penrose himself supported a hypothesis called “cosmic censorship” that no one can ever empirically observe a “naked” singularity: it must be “clothed” or surrounded by an event horizon to prevent one from seeing it. So, Penrose not only accepts but promotes the idea that a singularity is not empirically observable. So, how exactly is this science? It does not seem to be.

Therefore, it is all important to understand the exact implication (“interpretation”) of an unobservable mathematical infinity called a singularity. Does it necessarily imply the formation of a black hole?

Let us first try to understand Penrose’s version of the story, repeated in the Nobel prize citation. By way of background, a black hole, or the Schwarzschild solution, was the first solution to the equations of general relativity. Amusingly, the solution was found even before those equations were finalised. But, until the 1960’s no one was sure whether black holes would actually form as a result of gravitational collapse. For example, in Newtonian physics, a rotating star (or the whole cosmos, if it rotates) would speed up its rotation as it collapsed and become smaller, because angular momentum would be conserved. This is similar to the way a figure skater speeds up her rotation by closing her arms and slows it down by spreading them. Consequently, it (the collapsing star or cosmos) would ultimately explode back on Newtonian gravitation. Another doubt was that the Schwarzschild solution assumes perfect spherical symmetry, and no one was sure whether or not slight departures from spherical symmetry, as were likely to occur due to perturbations, could prevent the formation of a black hole during gravitational collapse.

This was the issue that Penrose’s singularity theory (on his version) originally claimed to have tackled. And this is what the citation by the Nobel prize committee means by claiming that Penrose showed that black holes are a robust consequence of general relativity: that rotation or small departures from spherical symmetry will not prevent the formation of a black hole. But does that Nobel citation do anything more than repeat an old story?

The formal mathematics of singularities

Thus, in my 1997 debate with Roger Penrose, I challenged also the mathematical aspects of that story that a singularity somehow means the formation of a black hole. Considering that a singularity has no clear-cut empirical consequences, we first need to understand more precisely what a mathematical singularity is. A singularity involves a mathematical infinity of some sort.

More technically. a singularity means geodesics cannot be extended. (Geodesics, by the way, are extremal paths.) But what does that mean? That can only be explained in a round-about way. First we need to understand what a geodesic in space-time is in physical terms. A null geodesic is the path of a light ray. A time-like geodesic is something like the path of a material particle, but not quite. The related simplifying concept of a point mass or test particle was used heavily in Newtonian physics, but it presents unresolvable difficulties in general relativity. A point-like test particle of finite mass would already be a black hole. (As an aside, the absence of a decent specification of matter in general relativity is just because the theory was formulated by a mathematician—Hilbert—who cared about the mathematics, and especially the geometry, more than the physics.) General relativity forces us to think in geometrical terms about geodesics in space-time, not particles of matter or their trajectories.
Anyway, the “accepted” workaround to the problem of no-point-particles in general relativity, is the geodesic hypothesis, that a time-like geodesic can be interpreted as the world-line (past and future history) of a test particle with “infinitesimal” mass, whatever that term “infinitesimal” might mean (if anything). We must accept this hypothesis because it is socially accepted, and science these days, especially the science of things far away and not directly observable, is increasingly about social acceptance and uncritical trust in experts.

Accordingly, a singularity, or the inability to extend time-like geodesics, was interpreted as the beginning or end of time for a test particle: a moment of creation or destruction. But since classical general relativity does not allow (or is believed not to allow) for the creation or destruction of matter, so a singularity was also interpreted as a moment when the equations of general relativity break down.

A singularity, as defined above, is a geometrical notion. What I pointed out in my debate with Penrose was that the failure of geometry is not synonymous with the failure of physics: a “singularity” does NOT necessarily mean a failure of the equations of general relativity (“laws of nature”). This requires a long excursus for the layperson, although I could explain it compactly to Penrose.

Failure of physics or failure of math?

First of all, a singularity is a very common occurrence in physics, a characteristic feature of (hyperbolic) partial differential equations. Like geodesics, the so-called characteristics of a partial differential equation may converge to give rise to a singularity. This happens, for example, in a fluid, in the most mundane situations when a firecracker bursts in air, or with the breaking of a long sea-wave, as it approaches a sea shore.

Nobody talks of God, creation, or even a black hole in the context of these mundane singularities. Instead one speaks of the breakdown of smoothness: the explosion of a firecracker results in a rapidly expanding blast wave at the front of which a sudden or discontinuous change takes place in the air pressure, temperature etc. That is,

smooth

solutions of the equations of physics cannot be extended beyond a singularity.

The equations of physics, being differential equations, seem to break down at such a singularity because a non-smooth function cannot be differentiated (on the college-calculus understanding of differentiation). Or to put it another way, any attempt to differentiate such a function could result in a mathematical infinity of some sort.
Nevertheless, in the everyday physics of fluids we do NOT speak of a breakdown of physics, or of God, or creation at such a “singularity”. First we do not need to go by mere trust in an “expert” on what such a singularity means, for we can actually see what happens at such a singularity: nothing much, a bit of froth in the case of a breaking wave, a bit of noise in the case of a firecracker. What we observe is that a surface (or, more technically, a hypersurface) of discontinuity forms: the blast from the fire cracker is felt by the ears as a sudden (discontinuous) change in pressure etc. It is only the continuous or smooth solution of the fluid equations that cannot be extended beyond such a singularity. A shock wave develops.

Secondly, in fluid dynamics, there is a backup theory to cover for the failure of the differential equations or “laws of physics”. The fluid is regarded as a continuum but only as an approximation: quantities such as pressure, density etc, of the fluid (such as air, water) are statistical averages. They are not present at the molecular level. But across a shock wave, the changes in pressure, density etc. take place in a very tiny distance: of the order of the mean free path of molecules. Statistical averages (involving a large number of molecules) cannot be validly taken over such tiny distances, so the continuum approximation breaks down at a shock wave. Hence, the shock can be theoretically regarded as a (hyper)surface of discontinuity. (We speak of a hypersurface, since a shock wave evolves in time.)

So, all that happens at such a “normal” singularity is that the continuum approximation breaks down, but we can still do physics, though we cannot continue to use the college-calculus understanding of the differential equations. But in general relativity, a mathematician’s theory, there is no proper description of matter. Though one knows that matter consists of discrete particles, from electrons and protons to stars, in classical general relativity one is forced to speak of a fluid since this crude description of matter is the only one that fits in the theory, the fit being by analogy with fluid dynamics. But there is no general relativistic statistical mechanics of a gas in a box, so in general relativity there is no other physics to fall back upon when the continuum description of matter as a fluid fails. Though the analogy with fluid dynamics has no real basis, the breakdown of the continuum approximation, in general relativity, could easily be wrongly interpreted as the breakdown of the theory or the breakdown of the “laws of physics” exactly as Hawking did, in claiming that general relativity proves its own failure.

This is simply wrong, though few dare say it, because in such matters, science is all about “proof by authority”, especially where complicated math is involved.

Conservation laws

To understand why this is wrong, let us again ask the question regarding the meaning of a singularity, but in a more mathematical way. How does one mathematically handle the problem of a firecracker or blast wave or shock wave in classical fluid dynamics, within the continuum formulation (without resorting to statistical mechanics, which resort is not available within general relativity)? How does one mathematical handle the related problem of infinities? This problem arises in the theory of blast waves from a bomb, such as a nuclear bomb, or the design of supersonic aircraft etc.

The classical answer is that one abandons the formulation of the equations of physics as differential equations and goes over to the integral formulation of conservation laws. For example, a point mass results in a singularity even in Newtonian physics, but this singularity is not of any particular consequence, for the integral form of the conservation laws would still hold in any small volume surrounding the point mass. (On the so-called fundamental theorem of calculus, differentiation and integration are inverse processes, though the theorem might fail as we change the definitions of the derivative and the integral, for example, by switching to the Schwartz derivative or the Lebesgue integral.)

However, viewed from a purely mathematical perspective the matter of shifting from differential to integral equations is NOT trivial for reasons that few (mathematicians or physicists) understand.

Riemann’s mistake

Thus, the celebrated mathematician B. Riemann (of Riemann-hypothesis fame) erred on this question of discontinuities or shock waves in classical fluid dynamics.
He erroneously assumed the conservation of mass, momentum and entropy, whereas entropy is not conserved across a shock but increases. (Later, the correct shock conditions—for a “perfect” fluid—were obtained by Rankine and Hugoniot.) For the smooth flow of a fluid, entropy is conserved, and the equations of conservation of mass, momentum, and energy are equivalent to and interchangeable with the equations of conservation of mass, momentum, and entropy. But this is NOT the case for non-smooth flows, for deep-seated reasons. That is, the two ways to formulate the “laws of physics” using differential equations and integral equations are NOT equivalent in the case of non-smooth flows even in classical fluid dynamics. In Einstein’s language one might say "God fumbles about the laws of nature at a discontinuity."In general relativity, there is a bigger problem, since there is no clear-cut integral formulation at all!

Products with the Dirac delta

So, a singularity can be seen as a mathematical problem regarding the understanding of the infinities thrown up by an application of a wrong understanding of calculus to physics. Forty years ago, I addressed this problem as a preliminary part of my PhD thesis at the Indian Statistical Institute, Kolkata. Of course, at that time I was very much into formal mathematics, so the solution I gave was from within formal mathematics.
Briefly, on the college-calculus, differential equations fail at a discontinuity because a discontinuous function cannot be differentiated. However, by the end of the 19th c., Oliver Heaviside was not only differentiating discontinuous functions, he was using this as a technique to solve differential equations in electrical engineering. The Heaviside function is a simple step function of one variable, which is zero when its argument is negative, and one when its argument is positive, so it has a simple discontinuity at zero. Later, Paul Dirac, originally an electrical engineer, took this up, and the derivative of the discontinuous Heaviside function came to be known as the Dirac delta function. That delta function is “infinite at zero and infinitesimal elsewhere”. (That is, it is zero “almost everywhere”. Nevertheless, its (Lebesgue) integral is not 0 but 1.)

The European (mis)understanding of calculus, using the continuum, brought the delta function into disrepute, but its use seemed unavoidable in quantum field theory. Later, the work of the Russian mathematician Sobolev, the French mathematician Laurent Schwartz, and the Polish mathematician Jan Mikusinski gave (different) understandings of the delta function. Of these, the Schwartz theory became the most popular (and formal mathematics is entirely about social consensus).

But though the Schwartz theory made the discontinuous (hence non-differentiable) Heaviside function infinitely differentiable, it was a linear theory (due to a result called the Schwartz impossibility theorem): the Dirac delta function could not be multiplied with itself. Therefore, one could not directly use the Schwartz theory to understand discontinuous solutions by changing the meaning of the derivative at a discontinuity in the non-linear differential equations of physics, such as the equations of fluid dynamics or general relativity. To cut a long story short, I solved this problem of defining products of Schwartz distributions in my PhD thesis, using something called Nonstandard Analysis which allows one to define and use infinities and infinitesimals at an intermediate stage within formal mathematics.

Many definitions of the products of Schwartz distributions have been given. But my solution to this problem is unique in that it allowed the equations of physics to hold AT a discontinuity. For example, the equations of physics at a shock wave, using my definition of the product of Schwartz distribution, are just the Rankine-Hugoniot equations. [26] At a star boundary, using the equations of general relativity, they are the classical conditions of Papapetrou etc. [27] At a general relativistic shock wave (in a perfect fluid) they are just the Taub conditions, at a surface layer or thin shell of matter, they are the Israel conditions, and so on. The most general conditions for relativistic and non-relativistic shocks, surface layers etc. were derived by me, [28] using this process, and they obviously go beyond all earlier conditions, including Taub’s conditions for relativistic shocks which apply only to perfect fluids (which do not exist in reality, since all real fluids, such as air or water are imperfect; for example, they have some viscosity and thermal conductivity).

The immediately relevant point here is this: on my analysis, a surface layer must accompany a shock in a real fluid. In the case of general relativistic shocks, on this (hyper)surface the geometries on the two sides are not the same: the hypersurface inherits different (extrinsic) geometries on its two sides. Therefore, there is no unique way to extend a geodesic across such a surface. To cut a long story short, the equations of relativity will continue to hold at such a singularity even though geometry fails. (There can, in principle, be more drastic situations, of a gravitational screen, [29] where the surface does not even have a unique intrinsic geometry, but we won’t go into that.) This failure of geometry happens even in the case of a simple shock wave in general relativity. In brief, what fails at a singularity is only geometry, not necessarily physics, and this happens even in the case of shock wave in general relativity.

This argument is fatal to Penrose’s thesis, for there is no way in Penrose’s theory to understand exactly what sort of singularity develops, and there is no way in his theory to do physics when geometry fails. Instead of a black hole, the singularity might refer to just a common shock wave, for example, which might developed during the course of gravitational collapse. As I put it in my book, Eleven Pictures of Time, the singularity-God may be just a firecracker. Classical differential geometry fails at a singularity. But that is insufficient knowledge to classify a singularity precisely, or to infer the existence of a black hole as a consequence purely of general relativity as the Nobel prize committee has erred in doing.

Ignorance of math among singularity theorists

All this is not widely known, because science today is about Western endorsement, and almost no one in the West seems to have understood my work on mathematical general relativity (which is good in a way since no prominent mathematician plagiarised it the way my other work was systematically plagiarised, e.g. as Michael Atiyah plagiarised [30] my critique of Einstein!).

In fact, few even among singularity theorists understand the mathematics. Some are unfamiliar even with the Schwartz theory. Thus, at a 1986 Stockholm conference on general relativity (GR11). I heard a plenary lecture by the same F. J. Tipler who spoke of quantum mechanical states represented by the Dirac delta function. I raised the elementary objection that quantum mechanical states were vectors in a Hilbert space, modeled as the space of (Lebesgue) square integrable functions, and the Dirac delta function was not square integrable, and, in fact, its square was not even easily defined. Tipler had no answer to my question, and hemmed and hawed and hawed and hemmed for a whole long minute on the stage. So excruciatingly long, in fact, that even I felt embarrassed for having asked the difficult question, and gave him a loophole by suggesting that he might mean “rigged Hilbert spaces”, a suggestion which he grabbed like a dying man grabs a straw, obviously without knowing what a rigged Hilbert space is, thus losing all future credibility with me.

This issue (“what kind of singularity”) was also independently analysed from another perspective quite deeply by C. J. S. Clarke, [31] who later decided to leave relativity to serve the church, the last time I met him and stayed with him in Southampton, while giving a talk in the math department there, twenty years ago. However, Clarke, in his other published work [32] made the mistake of assuming that the Colombeau product [33] could be used in the manner of my product for an extended understanding of the equations of general relativity. This way of extending classical physics to discontinuities is unique with my product and does not apply to various other products, such as the Colombeau product or the Hahn-Banach product used in renormalisation in quantum field theory.

Penrose, a mathematician, perhaps understood the strength of my objection; I cannot judge, for he failed to respond to it, and was deceptive in other ways. My objection creates a major problem for the geometric approach to general relativity, for the correct form of the equations of general relativity at even a common singularity such as a shock wave may have to be decided empirically, and not by authority, as we saw in the case of Riemann’s error. On my approach, the problem that two differential equations with equivalent smooth solutions may have inequivalent singular solutions is seen as due to the failure of the associative law for my product of Schwartz distributions. Therefore, the nature of a singularity cannot be decided in a purely geometric way, and without empirical inputs, as I emphasize in my later book Cultural Foundations of Mathematics. This conclusively shows that Penrose’s interpretation of a singularity as necessarily indicating a black hole is not correct.

The Nobel prize committee, by endorsing singularity theory, has either erred or it has made a political decision to promote a way of using science which has been extensively used to argue in favour of Christian theology. It would be naive to imagine that it would pass up the opportunity to do religious politics through science.

As we will see in the next part, there is a more recent aspect of the politics arising from the debate on decolonisation of math in South Africa, where I clashed with Stephen Hawking’s co-author, Ellis on the issue of singularity theory.

Part 3: Singularity theory and the decolonisation of math 

The first part of this article explained how a singularity in general relativity was peddled as proof of the validity of Judeo-Christian theology as physics, brazenly by Tipler and more subtly by Stephen Hawking and G. F. R. Ellis. The second part of the article explained that, regardless of the unjustified assumptions about time made by Penrose, Hawking et al, assumptions about time central to post-Nicene theology, and needed to prove the existence of a singularity, what fails at a singularity is only the college-calculus understanding of the differential equations of general relativity. With my modification of the calculus (“product of Schwartz distributions”), within formal mathematics, the differential equations continue to make sense even when the college calculus fails. The conclusion: what fails at a singularity is only geometry not necessarily physics. But this is fatal to Penrose’s and (Hawking’s) geometric methods of proving the existence of a singularity, for those methods cannot distinguish between the failure of physics and the failure of differential geometry.

Subsequent to my 1997 debate with Penrose, a major new development has taken place: it has been demonstrated that the calculus originated in India and was transmitted to Europe. [34] Further, as often happens in cases of those copying from others, Europeans failed to properly understand some finer points of what they copied. This did not deter Europeans from building extremely untruthful tall tales of the achievements of Newton and Leibniz regarding the calculus, tales based on the religious and legal dogma that the discoverer is the first Christian to spot the ideas. (Those who don’t learn from history are condemned to repeat it: the same process of intellectual appropriation is repeating in the present tense since the very story that the calculus was stolen was itself serially plagiarized along similar “principles”! [35])

My epistemic test is a good way to expose such false history: those who claim or are given false credit, like Einstein for relativity or Newton for calculus, often failed to understand the very thing they are credited for. Specifically, Europeans failed to understand how to sum the infinite series of the Indian calculus. [36] After centuries, they did evolve some sort of a solution inventing the continuum (or formal real numbers), which invention was incomplete until the invention of formal set theory in the 1930s. Exposing this monumental and continuing theft of history is important to understand that science (including the experimental method [37]) developed in the non-West thousands of years before it developed in the West, so it is not necessary for the colonized to imitate the colonizer.

However, the issue right now concerns philosophy not history. By now, an alternative understanding of calculus has been established, and widely taught, based on the way the calculus originated in India. [38] Contrary to the myth about it, mathematics is NOT universal: the Indian way of doing the calculus was different. Accordingly, a new issue has arisen regarding singularity theory. Is the failure of college calculus at a singularity just due to difficulties peculiar to the Western (mis-)understanding of the Indian calculus?

The challenge to formal mathematics

At its core, the different way of doing calculus extends far beyond just the calculus, to a different way to do mathematics, and challenges the entire formalist philosophy of mathematics, used by Penrose. The question is not who did calculus first; the question is how should it be better done today? This question also relates to a major new political development: the attempts to decolonise mathematics across the last decade, for the calculus is a major stumbling block in the teaching of mathematics today.
The claim of cultural superiority was a key enabler of colonialism and colonial education [39] just as the claim of racial and religious superiority enabled the Christian [40] “moral justification” of physical genocide of indigenous populations (in 3 continents) and slavery and apartheid in a fourth.

As such, it is great blow to that imagined colonial "superiority" that not only was the calculus (needed for science) not invented in the West, as false Western history (mis)informed us for centuries, it was not even properly understood in the West and needs to be reformed today. [41] An even bigger blow to Western vanity and the Western grip on the colonised mind is this: the philosophy of formal/Western/colonial mathematics, globalized by colonial education, involves church myths and superstitions, which are poisonous and need to be expunged from our education system. This attempt to retrieve real mathematics from the mess that the West has made of it strikes at the roots of colonial dominance (which continues even into “post-colonial” times), for mathematics and the calculus is what is needed for science, and science is the key reason for continuing colonial education.

Church myths and superstitions in formal mathematics

To start with, it is a church myth that formal mathematics (or the “axiomatic method”) originated from Greeks, and, in particular, a Greek called Euclid (“aqi-des=rational geometry” [42]). Actually, my prize of Rs 2 lakhs for primary evidence for Euclid [43] has been standing unclaimed for over 10 years. Nearly 2 decades ago David Fowler [44] the leading Western historian of Greek mathematics, familiar with primary sources of Greek mathematics, admitted the lack of any primary evidence for “Euclid”.

Despite the lack of evidence, many people feel they can hang onto the church myth about the origins of mathematics. They say “the book is there”. They foolishly believe that the myth about the late 9th c. book (that it has axiomatic proofs) must be true, and many people in so many centuries must have read the book carefully. They grossly underestimate the church capacity to propagate brazen untruths. Thus, since around 1125 when the book first came to Europe, for some 750 years until Dedekind, no one carefully read the book. Instead ALL Western scholars just went by the myth about the book and failed to notice that even its first proposition lack an axiomatic proof.
It is absolutely laughable that the University of Cambridge incorporated this foolish myth (about axiomatic proofs in the book) in its revised exam regulations even as late as 1893. [45] Alas, by the end of the19th century the West was forced to admit the falsehood of the myth: that fact is that there isn’t a single axiomatic proof in the Elements, the book attributed to Euclid. In fact, it is quite obviously a book related to the Egyptian mystery geometry championed by Pythagoras and Plato. The commentator Proclus explained this explicitly. [46] Again, for example, the book has diagrams which aid in mathesis according to Plato (Phaedo) but are irrelevant to formal proof as noted by Russell.

Eliminating the myth of Euclid uncovers the face of the church: that axiomatic reasoning (reasoning without facts) was a form of reasoning invented by the crusading church (not “Euclid”) in the 12th century in support of its Christian rational theology. Mere reason is NOT against superstition. Axiomatic reasoning, which avoids facts, enables one to use reason to prove absolutely anything that one wants. Rationalists are yet to understand how the church doublespeak about reason is used to fool them and how they still confound normal reason (reason plus facts) with formal reason (church reason minus facts, axiomatic reasoning). This church method of reasoning enabled Aquinas to reason about angels [47] which do not exist in fact.

This trick of assuming whatever one wants to prove what one wants to prove was also the method used by Hawking and Ellis in using their chronology condition (an assumption critical to the post-Nicene church politics of inequity) to prove the existence of a singularity which they interpreted as a moment of creation.

Formal mathematics is based not on the text attributed to “Euclid” but on its fraudulent church reinterpretation to suit this theology of reason (without facts). Westerners, like the colonized, hegemonized by the church, were simply unable to see the difference between reason convenient to the church and reason (based on facts) hostile to the church. They never asked how mathematics and theology could both be based on reason, or could both use the same text (“Euclid”) for so many centuries.
In this hilarious comedy of the birth of formal mathematics, the last act takes the cake. In the 20th century it was admitted that there are no axiomatic proofs in the Elements. Russell wrote that it was “no less than a scandal that he [Euclid] should be taught to boys in England”. [48] (We still teach it!) The 20th c. rewrite of that church text by Hilbert [49] et al. accepted that “Euclid’s” book lacks deductive proof of even its first proposition, or its penultimate, the “Pythagorean theorem”. But it is fallacious to imagine that people who reject one error reject all. Thus, both Russell and Hilbert stuck to the superstition that pure deductive proofs are somehow superior since "infallible".

Real formal mathematics, starting from Hilbert and Russell, is founded on the continuing church superstition that pure deduction (=reasoning minus facts) is infallible. That this is a superstition [50] should be obvious to any maths teacher who has painfully had to correct the numerous wrong proofs that students submit in exams. This superstition (of the infallibility of deduction) is particularly laughable from the Indian perspective where the Lokayata rejected deduction as extra-fallible, thousands of years before the church declared it to be infallible! There is also no shortage of well-known mathematicians, such as Kosambi, Atiyah, etc. who have made mistakes. And it will not do to claim that valid deductive proofs are infallible, because so are valid empirical proofs, by definition! The difficulty is about knowing whether a given proof is valid.

Can this state of affairs (of science based on mathematics based on superstition) be corrected?

The consequences of colonial education

The first barrier is ignorance. Though colonial education supposedly came for the sake of science most people entirely overlook the actual consequences. The fact is that after nearly 2 centuries of colonial education the net result is (a) widespread mathematical illiteracy and (b) belief in all sorts of superstitions and myths about mathematics.
A typical such belief is that mathematics is universal and cannot be decolonized. People typically ask "how can 1+1 = 2 be decolonized?" They have in mind the simple empirical proof of 1+1 = 2 which is not admissible in formal mathematics. To explain the complexities, at a recent talk in JNU, I reiterated my "Cape Town challenge" for the professors of JNU. The challenge is to prove that 1+1 = 2, in formal real numbers (or the continuum). (Note that 1 as a natural number is different from 1 as a "real" number; Peano’s axioms do not apply to the latter.) “Real” numbers are needed for the Western understanding of calculus, and taught in school from class IX onwards. The proof of 1+1=2 is required from first principles without assuming any result of formal set theory, and developing all the intermediate results required directly from the axioms of set theory, in the manner of Russell’s 378 proof of 1+1 = 2 in cardinals in his Principia Mathematica. As can be checked from the video of the JNU lecture [51] I offered a reward of ₹10 lakh for a such a proof of 1+1=2 within 24 hours, and a reduced reward of ₹1 lakh for a proof within a week. No one claimed the reward. This is the extent of the mathematical illiteracy spread by colonial education that even the professors of mathematics education in our top universities do not know why1+1=2. The situation is not very different in IITs [52] or our other universities.

This total mathematical illiteracy among the colonially educated is combined with two deep-seated superstitions: (a) the superstition (most manifest in Wikipedia) that the West and only the West is trustworthy, and (b) the belief that any change from blindly imitating the West can only be for the worse. (“Doomsday awaits the unbelievers.") The colonized hence resist change.

For example, a stock argument of the ignorant against change, and in favour of current math, is that “it works”. But what exactly works? The ignorant don’t understand how rocket trajectories, for example, are calculated. They conflate normal and formal math, the way rationalists conflate normal reason (reason plus facts) with formal or church reason (reason minus facts, faith-based reason). [53] What works (and works better) is NOT the formal mathematics of proof but the normal mathematics of calculation (much of it imported by Europe from India for its practical value, starting from elementary arithmetic algorithms). For a concrete demonstration of how normal math works even when formal math fails, see the point about stochastic differential equations (from twenty years ago [54]) repeated in this recent video [55] of the talk on statistics for social sciences and humanities at JNU. A simple rule of the thumb is that anything which can be done on a computer (such as calculation of rocket trajectories) is normal mathematics, and most practical applications of math today involve computers.
The bigger problem is this: from this position of the darkest ignorance wrapped in the deepest superstition, even discussing an alternative to Western ethnomathematics is taboo for the colonized.

Censorship

If someone does not like this critique of current math teaching or practice, they are welcome to debate it publicly. I have been inviting people for such a public debate for several years. [56] However, such a public debate on the philosophy or teaching of math has proved to be impossible over the last decade across several countries. Every single mathematician or math teacher I know refuses to discuss the matter publicly.

Thus, the Malaysian Mathematical Society invited then uninvited me in 2011 as did JNU in 2013. Rohit Dhankar of Azim Premji University responded abusively to my point about an alternative philosophy of math on the principle that the fouler the abuse the more profound the philosophy. But when invited to a recorded debate he said he knew only school math (calculus is part of school math), but later chickened out of even a debate on school geometry. In 2018, the Palestine Technical University invited me to give two keynote addresses on decolonisation of math, but this was censored by Israel which refused to grant me a visa. [57]

Censorship is the classic devise to preserve superstitions, a device invented by the church to facilitate its rule by superstitions. But why censorship in math? Is there something obscene [58] about the possibility of an alternative philosophy of math? Is there something obscene in speaking about Western ethnomathematics or the peculiar way in which mathematics developed in the West under the overarching influence of the church?

In this context, what happened in South Africa during the Rhodes Must Fall agitation was interesting. In 2016, I gave a talk at the University of South Africa on decolonizing mathematics which possibility unsettled the White South Africans steeped in apartheid. One Karen Brodie, dishonestly pretending to take the side of the colonized, explained an “alternative” way. She said that Blacks and women are bad at mathematics, and the right way to “decolonize” math was to teach them to imitate the thinking of the white males who, she said, had created the subject of mathematics. This is of course the classic strategy of “deep colonization” (not decolonization) used by the church and colonialism: “we are superior, imitate us”. I objected in a 1000 word article published in the Conversation, “To decolonise math stand up to its false history and bad philosophy”. [59] The article went viral and was reproduced worldwide, including by Scroll and the Wire [60] in India.

But it was later censored by the South Africa editor of the Conversation. The reason? On racist South African editorial norms, even after apartheid, non-Whites are considered unreliable and hence not permitted to state their own thoughts and not permitted to cite their own prior published works, such as my book Cultural Foundations of Mathematics (Pearson Longman, 2007). They can only imitate Whites and must avoid any novel thoughts. Apartheid was possible because racists could hence advance the most idiotic claims of superiority without fear of being challenged, exactly like the medieval church advanced its utterly fanciful claims like virgin birth and defended them by censoring anything to the contrary, as Wikipedia still does by declaring it “unreliable” and simply burying it.

One such fanciful claim is that advanced by Brodie that mathematics was created by dead White men, as is also depicted through images in our current school texts. This belief is based on the churchist and racist history of “Greek” origins. [61] To counter this, I argued in the censored article, [62] first, that "Euclid" was actually a black woman. [63] The second argument which enraged the racists was that the West and Whites were inferior in mathematics, even elementary arithmetic, and (early Greeks and Romans, and Europeans up to the 16th c.) were thousands of years behind black Egyptians. They, hence, could NOT have invented mathematics. This is clear from the non-textual evidence of numerals (absence of fractions in Greek and Roman numerals) and the related issue of the (still defective) Roman calendar. Fractions are found in the Ahmes papyrus, [64] but were first introduced into the Jesuit syllabus (from India by Clavius) only in the 1570’s, so late that the Gregorian calendar reform of 1582 still used the crude device of leap years instead of stating the duration of the year as a precise fraction. It is important to focus on such non-textual evidence because it cannot be easily manipulated the way church priests manipulated and forged and mistranslated texts, apart from wildly mis-attributing all sorts of things to “Greeks” based on excessively late sources.

One cannot blame churchists and racists alone, for the censorship of my article happened not only in South Africa, but throughout the world, and in India as well. Both the Scroll and the Wire first reproduced it then took down my article (though Wire eventually put it back). The Scroll indirectly declared that my article was removed since it involved “bad mathematics”. But when asked to specify what was the “bad mathematics” the Scroll editor had no answer, and lamely said he would use instead the euphemism “academic debate” though the debate was exactly what was being avoided [65] by censorship!

The key point to understand is that the church superstitions in mathematics are preserved by using exactly the church tactic of systematically avoiding any open debate or public discussion about them.

It was against this background that I visited the University of Cape Town (UCT) in 2017 for a public debate on the decolonization of mathematics and science. [66] Recall that (a) Stephen Hawking’s co-author G. F. R. Ellis is in the mathematics department of UCT, and (b) the singularity theory of Hawking and Ellis is a great example of how science is used to assist church propaganda, [67] an example considered in detail in my book The Eleven Pictures of Time. Accordingly, singularity theory is a prime example of how science needs to be and can be decolonized.

This nexus between science and church is facilitated by axiomatic mathematics. It is one thing that axiomatic/formal/colonial math is based on the superstition that deduction is infallible, a superstition validly rejected by the Lokayata in India thousands of years before the church adopted it. It is another thing that axiomatic mathematics enables people to smuggle in politically convenient but metaphysical (empirically unverifiable) postulates such as the chronology condition of Hawking and Ellis, and then falsely claim, as e.g. Hawking and Tipler did, that their conclusions are an inevitable consequence of general relativity.

Accordingly in my advance abstract [68] for the UCT discussion I made what I considered was a minor change: I assumed that it would be possible to discuss the more mathematical and technical aspects of singularity theory separately in the mathematics department.

Now Ellis and his gang of church supporters (not all White) were well aware of my earlier debate with Penrose, and my long-published critique that the singularity God was perhaps merely a firecracker (as explained in part 2). They responded not in the academic manner which befits supposedly top-ranking academics, but in the typically scummy manner of the church (which is politically very influential in Africa). They threw mud at me, for they lacked valid arguments.

A campaign was started among mostly the White faculty in UCT to declare me a “conspiracy theorist”. Forget about evidence, no indication was ever given of what conspiracy theory I was accused of: for racists any nonsense accusation is proof of itself. The strategy of throwing mud ensures that there is no engagement with facts or arguments. (Because of the vast church reach this brazenly false allegation has been repeated subsequently even in US magazines like Undark from MIT, by an irresponsible journalist, and even by propagandists of the American Mathematical Society. [69]) On the same specious ground, the White faculty and their pawns petitioned the White Vice Chancellor not to allow a debate with me in the mathematics department. This is the desperate way in which Ellis justified avoiding open debate on singularity theory in the math department of the UCT.

If there was something obviously wrong in my arguments, Ellis could have just spent ten minutes exposing them. This would have taken far less effort than setting off a scurrilous campaign and mobilising political support to avoid a debate.
The real reasons for avoiding debate were threefold. First, Ellis lacked the mathematical competence to understand the technical complexities of my work even on the formal mathematics of singularity theory. [70] Obviously, as a local hero, he did not want his mathematical incompetence to be exposed, like Tipler’s was by me in Stockholm. The false propaganda misled many: even someone who wrote in my support [71] wrongly guessed that I had made some mathematical mistake, and gave the apology that I was a historian of science, not a mathematician or a philosopher of mathematics! Ellis correctly calculated that his mathematical incompetence would be saved from exposure by avoiding debate. Anyway, singularity theory is an example of how "top ranking science" survives on debate avoidance.

Secondly, as one of the originators of singularity theory, Ellis was well aware of its inherent weaknesses and did not want them to be exposed, especially not on his home turf. For example, the issue of the biased chronology protection postulate, and the patently wrong arguments given in its support by Hawking and Ellis. If the assumption is wrong obviously the whole theory falls to the ground. By debate avoidance, Ellis ensured that this weakness of singularity theory remained buried.

Thirdly, there is the issue of the church linkages. A debate on singularity theory would have exposed how easily religious dogmas can be slipped into “science”, through formal mathematics. This would have exposed Ellis’s bigger scam of using science to promote church dogmas. Of course, in this case the church linkages extend much deeper into the very foundations of formal/colonial mathematics.

Therefore, for good measure, Ellis set his student and fellow church goer, Murugan, also on the UCT math faculty, to attack my math decolonisation program thorough the racist South African press. Murugan told a series of lies on the Orosian belief that lies in favor of the church are rewarded by the Christian God, or that they would at least be rewarded by his racist bosses in UCT. For example, in the typical style of doomsday prophets, Murugan said my method of teaching calculus would result in Bantuization.

(Bantuization was a program of apartheid South Africa, during Ellis’s time, to teach black students the minimum they needed to perform hard labour.) It is easily checked that Murugun lied brazenly. My calculus course actually enables students to solve harder problems, such as the use of non-elementary elliptic functions, dropped as too hard from the usual calculus courses. This is the exact opposite of Bantuization. This is manifest from a cursory examination of even the tutorial sheets [72] of my calculus course from long before the Cape Town panel. Murugan is an example of how easily the mathematically illiterate colonised can be misled by a charlatan posing as a mathematical expert.

What is most striking about the whole episode is the parallel with church methods of defending the superstitions on which church power is based. Thus, the church invariably avoided debate with heretics, it often told brazen lies etc. Ironically, these techniques are being used to defend "science" which has been awarded the Nobel Prize!

Such scams in science are possible only because colonial education makes people ignorant of mathematics and science and also teaches them to uncritically trust Western authority. What people need to understand is that they can be cheated only by those they trust. And the Nobel prize is the ultimate and trusted “proof by Western authority” in science, for no empirical proof of a singularity will ever be forthcoming.

(Author: C. K. Raju, Indian Institute of Advanced Study, Rashtrapati Nivas, Shimla 171 005)


[1C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic/Springer, 1994) Fundamental theories of physics, volume 65.c

[2Thomas Aquinas, Sumnma Theologica, n.d., http://www.newadvent.org/summa/2091.htm.

[3Al-Ghazâlî, Tahâfut Al-Falâsifâ, trans. S.A. Kamali (Lahore: Pakistan Philosophical Congress, 1958); S. Bergh, Averroes’ Tahâfut al-Tahâfut (Incorporating al-Ghazâlî’s Tahafut al-Falasifa) Translated with Introduction and Notes, 2 vols (London: Luzac, 1969).

[4C. K. Raju, ‘Islam and Science’, in Islam and Multiculturalism: Islam, Modern Science, and Technology, ed. Asia-Europe Institute University of Malaya and Japan Organization for Islamic Area Studies Waseda University, 2013, 1—14, http://ckraju.net/hps-aiu/Islam-and-Science-kl-paper.pdf; C. K. Raju, ‘Response to Hoodbhoy’, Frontier Articles on Society & Politics, 11 April 2020, https://www.frontierweekly.com/views/apr-20/11-4-20-Comment-Response%20to%20Hoodbhoy.html

[5Statistics for Social Science and Humanities: Should We Teach It Using Normal Math or Formal Math?, 2020, https://www.youtube.com/watch?v=A9Og1k-Z5O4

[6C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003)

[7C. K. Raju, ‘The Christian Propaganda in Stephen Hawking’s Work’, DNA India, 16 January 2011, sec. Lifestyle, https://www.dnaindia.com/lifestyle/review-the-christian-propaganda-in-stephen-hawking-s-work-1495047

[8University of Cape Town South Africa, Decolonising Science Panel Discussion: Part 1, 2017, https://www.youtube.com/watch?v=ckbzKfRIi6Q

[9C. K. Raju, Mathematics, Decolonisation and Censorship, 2017, https://kafila.online/2017/06/25/mathematics-and-censorship-c-k-raju/.

[10“Plagiarism by ex-president of the Royal Society. 1: The facts”. Part 2: The cover-up by the American Mathematical Society, 3: Lessons for decolonisation of math.

[11How Colonial Education Changed Our Math Teaching | C.K. Raju, 2020, https://www.youtube.com/watch?v=Rm6d-bUmmGg

[12C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’, Philosophy East and West 51, no. 3 (2001): 325—362, http://ckraju.net/papers/Hawaii.pdf

[13C. K. Raju, ‘Decolonising Mathematics’, AlterNation 25, no. 2 (2018): 12—43b, https://doi.org/10.29086/2519-5476/2018/v25n2a2

[14Decolonise Math = Eliminate the Myth, Fraud, and Superstition in Formal Math, 2020, https://www.youtube.com/watch?v=rvSbGFiCsO8

[15Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs chp. 5, In Einstein’s Shadow.

[16C. K. Raju, ‘Penrose’s Theory of the Mind: a Rebuttal’, The Matter of the Mind, 22—23 December, India International Centre, New Delhi, 1997. The issue of politics in singularity theory was also discussed at several informal meetings on “Science and society” at Delhi University prior to that.

[17S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, 1973.

[18F. J. Tipler, The Physics of Immortality: Modern Cosmology, God, and the resurrection of the dead, Macmillan, 1994.

[1919C. K. Raju, ‘Science, Reason, Superstition. 1: Religion and Geopolitics’, Frontier Articles on Society & Politics, 7 June 2020, https://www.frontierweekly.com/views/jun-20/7-6-20-Science%20reason%20and%20superstition-1.html

[20C. K. Raju, “The curse on ‘cyclic’ time”, in The Eleven Pictures of Time, Sage, 2003, chp. 2.

[22C. K. Raju, ‘Time Travel and the Reality of Spontaneity’, Foundations of Physics 36 (2006): 1099—1113; C. K. Raju, ‘Functional Differential Equations. 5: Time Travel and Life”’, Physics Education (India) 31, no. 4 (2015), http://www.physedu.in/uploads/publication/21/344/1.-Functional-differential-equations-5-Time-travel-and-life.pdf.

[23Raju, Time: Towards a Consistent Theory.

[24F. J. Tipler, ‘General Relativity, Thermodynamics, and the Poincaré Cycle’. Nature 280, no. 5719 (1 July 1979): 203—5. https://doi.org/10.1038/280203a0.

[25Indeed, it is a problem even to estimate the mass of a galaxy. This is done using Newtonian gravitation, which approximates general relativity because of the difficulty of solving the many-body problem in relativity. But it has long been known that Newtonian gravitation (which works very well for the solar system) prima facie fails for the galaxy. A new hypothesis was invented to save Newtonian gravitation: that this failure is due to dark matter. This seems like accumulation of hypotheses, for nobody could pinpoint that nature of dark matter or its peculiar distribution for the last 80 years. For a quick summary, see C. K. Raju, ‘Functional Differential Equations. 4: Retarded Gravitation’, Physics Education (India) 31, no. 2 (June 2015), http://www.physedu.in/uploads/publication/19/309/1-Functional-differential-equations-4-Retarded-gravitation-(2).pdf.

[26C. K. Raju, ‘Products and Compositions with the Dirac Delta Function’, J. Phys. A: Math. Gen. 15 (1982): 381—96.

[27C. K. Raju, ‘Junction Conditions in General Relativity’, Journal of Physics A: Mathematical and General 15 (1982): 1785—1797.

[28C. 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.

[29C. K. Raju and N. Dadhich, ‘Is Gravitational Screening Possible?’, in GR10 (Padova: Abstract in Proc. ed. B. Bertotti, 1983); C. K. Raju, ‘Gravitational Screening’, in IAGRG XII (Pune, 1983).

30http://ckraju.net/atiyah/atiyahcase.html.

[31C. J. S. Clarke, The analysis of space-time singularities. Cambridge University Press,
Cambridge, 1993.

[32C. J. S. Clarke, J. A. Vickers, and J. P. Wilson, ‘Generalized Functions and Distributional Curvature of Cosmic Strings’, Classical and Quantum Gravity 13, no. 9 (September 1996): 2485—2498, https://doi.org/10.1088/0264-9381/13/9/013.

[33My technique of using a product to extend the equations of classical physics to a discontinuity does NOT work with other ill-thought out (but authoritatively endorsed) products. It is obvious that the products used in quantum field theory cannot be applied to classical physics. It is less obvious in the case or products such as Colombeau’s product. I did once try to bring out this difference empirically in a joint paper which I wrote with J. F. Colombeau ca. 1987 but never published. However, there are excellent theoretical reasons why that Colombeau product won’t work in classical physics. Unlike my product, the associative law holds for that product. Therefore, when using it, there is no way to understand why different forms of the same differential equations (with equivalent smooth solutions) will lead to different physics in the case of a discontinuity, as in Riemann’s mistake. Incidentally, mine is the only product which applies to both quantum and classical physics. C. K. Raju, ‘On the Square of x-n’. Journal of Physics A: Mathematical and General 16 (1983): 3739—3753.

[34C. 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).

[35“Prof. Raju’s charge of plagiarism found correct: UK varsity warns lecturer” Hindustan Times, Bhopal, 8 Nov 2004, http://ckraju.net/press/2004/HT-8-Nov-2004.gif. “George Joseph serial plagiarist”, http://ckraju.net/blog/?p=132, http://ckraju.net/blog/?p=166, etc.

[36C. K. Raju, ‘Marx and Mathematics-1: Marx and the Calculus’, Frontier Weekly, 28 August 2020, https://www.frontierweekly.com/views/aug-20/28-8-20-Marx%20and%20mathematics-1.html; ‘2: “Discovery” of Calculus’, Frontier Weekly, 31 August 2020, https://www.frontierweekly.com/views/aug-20/31-8-20-Marx%20and%20mathematics-2.html; ‘The European Navigational Problem and the Dissemination of the Indian Calculus in Europe’, Frontier Weekly, 4 September 2020, https://www.frontierweekly.com/views/sep-20/4-9-20-Marx%20and%20mathematics-3.html; ‘4: The Epistemic Test’, Frontier Weekly, 8 September 2020, https://www.frontierweekly.com/views/sep-20/8-9-20-Marx%20and%20mathematics-4.html

[38C. K. Raju, ‘Decolonising Mathematics’, AlterNation 25, no. 2 (2018): 12—43b, https://doi.org/10.29086/2519-5476/2018/v25n2a2.

[39C. K. Raju, ‘Education and Counter-Revolution’, Frontier Weekly 46), no. 7, Aug 25-31 (2013), https://www.frontierweekly.com/archive/vol-number/vol/vol-46-2013-14/46-7/46-7-Decolonising%20Hard%20Sciences.html.

[40C. K. Raju, ‘The Meaning of Christian “Discovery”’, Frontier Weekly, January 2015.

[41Raju, ‘Decolonising mathematics’, Raju, ‘Marx and Mathematics. 4: The Epistemic Test’.

[42Martin Bernal, personal communication, as distinct from uqli-des=”key to geometry” in C. K. Raju, Is Science Western (Farsi), trans. S. A. Mirhosseini and Roya Kiyanfar (Tehran: Iran Universities Press, 2012).

[43Goodbye Euclid Part 1 (Universiti Sains Malaysia, Penang, 2010), https://www.youtube.com/watch?v=sEK1FCrLHjU, Part 2, https://www.youtube.com/watch?v=MFf5co3G3R8; Part 3 https://www.youtube.com/watch?v=zomZU949Cnw

[44http://mathforum.org/kb/thread.jspa?threadID=381990&;
messageID=1175734%#1175734, Historia Matematica discussion list, 10 Nov 2002.

[45See, note, http://ckraju.net/geometry/cambridge-note.html. H. M. Taylor, Euclid’s Elements of Geometry (Cambridge: Cambridge University Press, 1893).

[46C. K. Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars (Penang: Multiversity and Citizens International, 2012).

[47Thomas Aquinas, Sumnma Theologica, n.d., http://www.newadvent.org/summa/1052.htm#article3.

[48Bertrand Russell, ‘Mathematics and the Metaphysicians’, in Mysticism and Logic and Other Essays (London: Longmans, Green, and Co., 1918), 94—95.
49David Hilbert [1898], The Foundations of Geometry (The Open Court Publishing Co., La Salle, 1950).

[49David Hilbert [1898], The Foundations of Geometry (The Open Court Publishing Co., La Salle, 1950).

[50Raju, ‘Decolonising Mathematics’, 2018; Decolonise Math = Eliminate the Myth, Fraud, and Superstition in Formal Math, 2020, https://www.youtube.com/watch?v=rvSbGFiCsO8.

[51Statistics for Social Science and Humanities: Should We Teach It Using Normal Math or Formal Math?, 2020, https://www.youtube.com/watch?v=A9Og1k-Z5O4.

[52Idiots and IIT, http://ckraju.net/blog/?p=41.

[53For an account of the difference between formal and normal math, see C. K. Raju, ‘Decolonising Mathematics’, Alternation 25, no. 2 (2018): 12—43, https://doi.org/10.29086/2519-5476/2018/v25n2a2. Or see this video ; ‘Decolonise math = Eliminate the myth, fraud, and superstition in formal math’, https://www.youtube.com/watch?v=rvSbGFiCsO8.

[54Also pointed out in passing in C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’, Philosophy East and West 51, no. 3 (2001): 325—362, http://ckraju.net/papers/Hawaii.pdf.

[55"Statistics for social science and humanities: should we teach it using normal or formal math?” https://www.youtube.com/watch?v=A9Og1k-Z5O4. JNU webinar, 16 Sep 2020.

[56See the call of public debate and the sole recorded conversation with India’s leading formal mathematician, M. S. Raghunathan, at http://ckraju.net/issa/conversation-draft-minutes.html.

[57“Israel denies visa for talk on decolonisation exposing Einstein”, http://ckraju.net/blog/?p=157.

[58‘Mathematics, Decolonization and Censorship: C. K. Raju’, KAFILA , 25 June 2017, https://kafila.online/2017/06/25/mathematics-and-censorship-c-k-raju/.

[59The original article is that http://ckraju.net/blog/?p=117.

[62C. K. Raju, ‘Was Euclid A Black Woman? Sorting Through The False History And Bad Philosophy Of Mathematics | Science 2.0’, 24 October 2016, https://www.science20.com/the_conversation/was_euclid_a_black_woman_sorting_through_the_false_history_and_bad_philosophy_of_mathematics-180581.

[63“Was Euclid a black woman”, http://ckraju.net/blog/?p=189.

[64M. Clagett, Ancient Egyptian Science: A Source Book, Vol. 3 Ancient Egyptian Mathematics (American Philosophical Society, Philadelphia, 1999).

[65“The Scroll and racist censorship: an open letter to the Scroll Reader’s Editor”, http://ckraju.net/blog/?p=125. The Scroll website is registered in Cambridge, Mass., and is a spin off from the Amar Chitra Katha series.

[66University of Cape Town South Africa, Decolonising Science Panel Discussion: Part 1, 2017, https://www.youtube.com/watch?v=ckbzKfRIi6Q; Office for Inclusivity & Change UCT, UCT Panel Discussion on Decolonising Science, 2018, https://www.youtube.com/watch?v=vWdqR-z6jIc.

[67C. K. Raju, ‘The Christian Propaganda in Stephen Hawking’s Work’, DNA India, 16 January 2011, sec. Lifestyle, https://www.dnaindia.com/lifestyle/review-the-christian-propaganda-in-stephen-hawking-s-work-1495047. Archived and annotated version at “Hawking Singularities”, http://ckraju.net/blog/?p=50.

[68C. K. Raju, ‘Abstract: UCT Panel, Decolonising Science’, 2017, http://ckraju.net/papers/uct-panel-decolonising-science-ckr-summary.pdf.

[69Plagiarism by ex-President of the Royal Society. 1: the facts. http://ckraju.net/blog/?p=183, 2L The cover-up by the American Mathematical Society, http://ckraju.net/blog/?p=184, 3: Lessons for decolonisation of math, http://ckraju.net/blog/?p=185. This blog was written as material for prior reading for a talk to have been delivered in South Africa in Jan 2020, which I could not deliver since I was unwell.

[70C. K. Raju, ‘Junction Conditions in General Relativity’, Journal of Physics A: Mathematical and General 15 (1982): 1785—1797; ‘Distributional Matter Tensors in Relativity’, in Proceedings of the 5th Marcel Grossman Meeting on General Relavity, ed. D. Blair and M. J. Buckingham (World Scientific, 1989), 421—23; arxiv:0804.1998. ‘Renormalization and Shocks’ Appendix to Cultura Foundations of Mathematics (Pearson Education, 2007).

[71Adam Cooper, “Surely good scholarship means having our perspectives challenged”, Daily Maverick, 11 October 2017. https://www.dailymaverick.co.za/opinionista/2017-10-11-surely-good-scholarship-means-having-our-perspectives-challenged/

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