Infinity, math, physics, and metaphysics

Can physics be done without infinity as taught in math (real analysis) today? Someone demanded an explanation in an email sent to my son.  (I guess the Raju family has the same problem as the Bernoulli family in Europe!  :-) .)

Ordinarily, I would not have responded, for people ought not to demand an explanation by email without bothering to read or understand what I have already written. But, similar doubts were expressed by a young woman (with a PhD in functional analysis) who attended my talk in Ramallah. They may again arise in future. So, I decided to respond.

Infinity is metaphysics. Infinity relates to eternity, so that the Western concept of infinity in present-day math is saturated with the church metaphysics of eternity.

Ironically, the figure for infinity, ∞, is still shaped like a serpent coiled back on itself and eating its own tail, and is an old symbol of quasi-cyclic time.

The linkage of infinity to eternity led to the first creationist controversy: over the nature of eternity, not evolution. In the 6th c. John Philoponus objected to Proclus’ notion of eternity based on quasi-cyclic time. Philoponus’ problem was that if the cosmos is eternal (as Proclus conceived it) it would not be created. That creationist controversy is still going on.

For example, Stephen Hawking claimed the cosmos was created with a “singularity”. (A “singularity” is nothing but an infinity of some sort.)  He concluded his only serious scientific book by identifying the “singularity” with “the actual point of creation” where there is a breakdown of the “laws of physics”. This conclusion is pure metaphysics, for there is no way to check it empirically.

In his popular book, Hawking explained the point of this metaphysical conclusion: because the “laws of physics” break down at the “singularity”, that leaves God free to create the world of his choice. Note that this is in accordance with the Christian notion of one-time creation (and contrary to the Islamic notion of continuous creation, or the Buddhist notion of non-creation, or the “Hindu” notion of periodic creation and destruction). The church heavily promoted this “scientific proof” of the correctness of its (post-Nicene) Christian theology.

People may be suspicious of the church but they implicitly trust scientists today. And, though few  (perhaps 2 or 3 among the 1.25 billion in India) have read or understood Hawking’s scientific work, hundreds of millions of people strongly believe he is a great scientist. Such gullibility and implicit trust is bound to be exploited by the church, which is ever on the lookout for new ways of doing its propaganda. Few people are even aware that Hawking reached his conclusion by postulating his “chronology condition” which denies quasi-cyclic time, and does so using exactly the same bad argument that Augustine used against Origen,  and which argument is at the foundation of post-Nicene Christianity. So, what Hawking did was to use the metaphysics of infinity to promote the politics of the church, like Augustine.

As I have also explained in The Eleven Pictures of Time, even granting Hawking’s assumed chronology condition, the  infinity (or “singularity”) arises because a discontinuity develops in a function which needs to be differentiated in the partial differential equations of general relativity or “laws of physics”. Such a discontinuous change in air arises also in the common phenomenon of shock waves resulting from the explosion of a firecracker. Westerners ruled out firecrackers because they wrongly believed the metaphysics of infinity which said the continuum was needed for the calculus; so they postulated, as Hawking does, that “nature is continuous” (in fact continuously differentiable :-) !).

Now, for several years I taught real analysis to students and mathematically proved in class that a discontinuous function cannot be differentiated. I also taught advanced functional analysis (and topological vector spaces and the Schwartz theory according to which every Lebesgue integrable function can be differentiated). In the advanced class, I mathematically proved the exact opposite that a function with a simple discontinuity can be differentiated infinitely often (and the first derivative is the Dirac δ).

So, we have two mathematical theorems: (1) a discontinuous function cannot be differentiated, and (2) a discontinuous function can be differentiated! (And Westerners believed that mathematics gives us eternal and universal truths!)

I asked the young woman in Ramallah, whether, as a certified expert in functional analysis, she could tell me which of these two opposing theorems I should use. Her response was typical of a formal mathematician: “use what you like”. (So, eternal and universal truths are just “what you like”!?) But that is hardly possible.

First our concern is with mathematics as it applies to physics or the real world. “Do what you like” in math spills over to “do what you like” in  physics, and that is not a legitimate way to do physics. Physics has to conform to the empirical world, not to one’s likes and dislikes. To put it another way, physics  must be refutable. If, like Stephen Hawking, one passes off one’s theological or metaphysical fantasies as physics, that leads to bad physics. This is a classic example of how metaphysics creeps into physics (through the metaphysics of infinity).

Secondly, and even more hilariously, there is the simple fact that one cannot do as one likes because both methods fail! (a) Discontinuities cannot be differentiated on the real analysis definition of derivative as limits (b) Schwartz derivatives cannot be multiplied pointwise, unlike ordinary functions (no δ.δ). So, neither definition can be used for the differential equations of physics. (See,  e.g. my paper “Teaching mathematics with a different philosophy. 1: Formal mathematics as biased metaphysics”.) We cannot do what we like, because formal mathematicians do not have a definition of the derivative (whether the one with limits, or the Schwartz derivative) which is good enough for current physics.

So, the fact is that Western formal mathematicians are still clueless about a good definition of the derivative, so clueless that they are not even aware that there is a problem! As clueless as Descartes was about summing infinite series, or Newton was about his “fluxions”, when the Indian calculus first reached Europe. The situation persists 450 years later despite all sorts of false stories about the historical origins of the calculus. (Those were deliberate falsehoods, because top church functionaries like Clavius, Tycho Brahe or, later, Kepler were surely aware of this import.)

Thirdly, students are not allowed to do what they like in math! They are told they must study calculus with limits, and formal reals or anything else decided in the West. The object of teaching this mathematics is to make them blindly subservient to Western authority as all formal mathematicians I know are. They don’t do what they like but only what the West likes. It is a shame on mathematicians worldwide, and especially in India, that they are so subservient, that they are too frightened even to discuss a new philosophy of mathematics.

Anyway, reliance on authority invariably fails. As is well known, this college-text method of doing calculus failed in the context of the infinities of the renormalization problem of quantum field theory (qft). Formal mathematicians call it the problem of defining “products of distributions”, though it is actually a problem of defining the derivative correctly. Few, even among mathematicians, understand exactly what they do. So, what they have done is to pile on the metaphysics about infinity, through definitions, and propose several dozens of definitions of “products of Schwartz distributions”. Of course, there is an infinity of different ways to extract a finite part from infinity, so there are an infinity of possible definitions!

How should one choose the right definition?

Now I proposed a simple test 30 years ago: if the definition is good enough it should not apply to qft alone, but also to classical physics. Specifically, it should also be able to handle simple discontinuities in classical physics, such as shock waves in a fluid or in  general relativity (or Hawking’s singularities).

On this basis I proposed a new definition (in my PhD thesis, back in the days when I still believed in formal math). By a happy stroke of luck (or intuition) I used Non-Standard Analysis, which tackles infinity in a way somewhat similar to (but enormously more complicated than) the non-Archimedean numbers with which calculus developed in India. So, it readily fits into my new philosophy of math called zeroism, which handles “infinity” in the way of Nagarjuna’s sunyavada, and the way it was done by Indians who developed the calculus.

There are many popular techniques to handle infinities in qft (e.g., Bogoliubov-Shirkov, which involves what formal mathematicians call the Hahn-Banach method). There are also other  definitions such as Colombeau’s simplistic (but technically complicated) definition of the product of distributions. All of these failed my simple test. I tried to point this out in a joint paper with Colobeau long ago, but he refused to send it for publication!

Mine is the only definition which passes the test.  For my resolution of the renormalisation problem for all quantum field theories, and my new shock conditions, see the appendix of Cultural Foundations of Mathematics, on “Renormalization and shocks” or this old paper on “Distributional matter tensors in relativity” (which, however, has a new abstract, clarifying the need for the empirical in mathematics). More recently, my technique for handling the infinities of qft was also successfully applied to Maxwell’s equations, in a joint paper with my son, to tackle the problem of infinity in classical electrodynamics: the runaway solutions with radiative damping.

Hilariously,  however, because formal mathematicians only understand Western authority (formal math is all about aesthetics/spirituality remember?) that useless definition by Colombeau is still being promoted!

Finally, my position as explained even in newspaper answers M. Waters (googling for whom only reveals “muddy waters” :-) ). The practical applications of the calculus are still done the Aryabhata way: EVERY practical application of classical mechanics, such as sending a man to the moon, is done by numerically solving differential equations. And a computer does not really understand infinity as I have explained on many occasions. The floating point numbers on a computer do not even obey the “associative law” so they are surely different from formal reals.

Just as the church declared everything natural as sinful (to profit from sins), formal mathematicians declare everything practical as erroneous! Life would be simpler and better for millions (of students who find math difficult, for example) if we instead declared the church and formal mathematicians, both, as erroneous, as they surely are.

For more on the foundational relation of the church and math see my book “Euclid and Jesus” How and why the church changed mathematics and Christianity across two religious wars”.

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