Peace for the World

Peace for the World
First democratic leader of Justice the Godfather of the Sri Lankan Tamil Struggle: Honourable Samuel James Veluppillai Chelvanayakam

Sunday, April 9, 2017

The Science-Mathematics nexus

The effectiveness of mathematics in the sciences 

article_imageby Kumar David- 

The fallout from what I thought would be a simple piece ("Ramanujan, Hardy and the God debate") which appeared on 26 March has been nuclear. Chief offender, Gamini Kulatunga, has peppered me with e-mails while Rajan Philips and others have been less cruel and fired only a few volleys. There have been many contributions in web Comments (Siri Gamage, Lester, Dr M. Gonlaskorale, AVB, Edwin Rodrigo and others). I declare upfront that I cannot do justice to all for two reasons. First, I am not competent to comment on spiritually oriented topics such as the Stages of Understanding in Buddhism or alternate non-Western knowledge systems (sorry Edwin and Lester) and I am a layman on the brain "100 billion axons and 100 trillion connections" and "randomness of billions of neurons and trillions of synapses arranged as tiny networks" (sorry Dr MG, Lester and AVB). These are significant topics for readers to follow up as I intend to do. Secondly, within the limits of a regular column, I have to be restrained and not bite off more than can be chewed.

I have now declared the limits of this essay; it will only explore the questions: Does maths in itself discovers new truths about the physical world? Can it create scientific knowledge? Does its use in a scientific thesis (Calculus and Newton’s Laws, Maxwell’s equations and Electromagnetism, Riemannian geometry and General Relativity) create knowledge not contained in the scientific theory itself? I say NO, I am on the side of the naysayers. But I am in bad company.

Einstein is reported to have remarked: "How is it that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"

Eugene Wigner a Nobel Prize winner wrote a 1960 article with the title: ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’; so you get his drift.

Mario Livio described as a renowned astrophysicist and author of 400 scientific papers asks: "Is math an invention of the brain? Or does it exist in an abstract world, humans merely discovering its truths?"

As opposed to this there was a certain contrarian chap who intoned: "The question whether objective truth can be attributed to human thinking is not a question of theory; it is a practical question. Man must prove the truth – the reality and power, the this-sidedness of thinking - in practice. The dispute over the reality and non-reality of thinking that is isolated from practice is a purely scholastic question."

I declare I am on the side of the contrarian chap and would to go further to replace ‘purely scholastic’ with ‘utterly meaningless’. Regarding the Einstein quote above, unless it is a fabrication he is wrong or at least incomplete. A genius can sometimes be wrong; Einstein, Newton, Plato and Gautama Siddhartha are not divine; their contributions are profound, but since human they are fallible. Einstein goofed at the heart of general relativity, inserting a fudge factor politely called the cosmological constant lambda (?), which later he shamefacedly admitted was "my greatest blunder".

Maths in Science

Let’s get to the heart of the matter: "Do the powerful effects of mathematics in science mean that the maths itself creates new knowledge absent in the scientific thesis to which it is applied?" Mario Levi (ML) is agnostic on this. I quote from his Math: Discovered, Invented or Both in Nova, 13 April 2015.

"How is it possible that all the phenomena observed in classical electricity and magnetism can be explained by just four mathematical equations? James Clerk Maxwell showed in 1864 that the equations predicted that varying electric or magnetic fields generate certain waves - the familiar electromagnetic waves (light, radio, x-rays) - eventually detected by German physicist Heinrich Hertz in the 1880s".

ML is confused. Are Maxwell’s Equations (ME) physics or mathematics? Their sheer beauty which can banish Cleopatra from a bridal chamber must not seduce one to forget that the body within is physics, the raiment without radiant mathematics. It can be written in two forms, differential or integral.

Forgive me this one unavoidable paragraphs; I will get back to terra firma fast. The first equation (B = 0) rather flamboyantly says "There is no true magnetism", meaning magnetism is a product of electric currents. The second equation says the electric field emanating from a region depends on the net electric charges contained therein. The famous third equation

known as Faraday’s Law of Induction, better known to schoolboys as the flux-cutting rule, is about how electricity is made in generators. The final equation supplements the first and says how much magnetism is made by a given electric current. Nobody needs to follow the maths since, qualitatively, what is being said is pretty straightforward.

Next the maths is manipulated to the n-th degree to derive weird and wonderful results; radio, TV and mobile-phone reception, electrical surges and marvellous circuits and devices. But it is not the maths that underpins it all; the maths is a handmaiden to extract implications that are already there.

A resplendent garment worn by ME after 1905 is its Special Relativity extension. This version is used to say things about places (frames of reference) moving at very fast constant speeds, close to the speed of light, relative to each other. (If there is acceleration we are in deep shit and need general relativity which is not easy to understand or explain).

Levi backtracks and after much meandering concedes the contrarian chap’s point. "Personally, I believe that by asking whether mathematics is discovered or invented, we forget that it is an intricate combination of inventions and discoveries. I posit that humans invent mathematical concepts - numbers, shapes, sets, lines - by abstracting from the world around". We can forgive his circumlocution, but a materialist would not have needed so much obliqueness before getting to the point.

A similar case can be made about Newtonian gravitation and classical dynamics on the science side and Calculus on the maths side, or general relativity as physics and Riemannian geometry as maths, or the relations of quantum electrodynamics as maths serving science. The wave-particle duality and Heisenberg’s uncertainty principle in no way disrupt the materialist standpoint. But I concede the jury is still out - exclusively at the quantum level – on "photon entanglement" ("quantum non-locality", what Einstein termed "spooky action at a distance"). This has no parallel in the macro world; trying telling the judge that when you had sex with that minor you were suspended in a state of probabilistic superposition! I am also unable to accept the fad that the real world does not exist in the absence of a conscious observer.

Gödel’s incompleteness theorem

One of my interlocutors (Gamini) has suggested that I address Gödel’s Theorem. A summary of the famous theorem from Wikipedia, with small linguistic adjustments, follows. Kurt Gödel (1906-1979) was a 25 year fresh graduate of the University of Vienna when he formulated it.

1. If a system is consistent, it cannot be


2. The consistency of its axioms cannot be proven within a system

The concepts are pretty self-evident – rigorous formal proof is not. The axioms of Euclid’s geometry cannot be proved within geometry (Gödel’s terrain was number theory). Now what is the relevance of all this to our science-maths concerns? The relevance is that after a scientific revolution, in passing causing one maths to be thrown out and another to find favour, then the former’s axioms too go out of the window. Euclid’s axioms lost out when general relativity lent on Riemann; pairing (quarks and leptons must exist in pairs) in the Standard Model of elementary particles is indeed a strange coupling-axiom; the axioms of quantum electrodynamics are an even odder genre. Maths is a handmaiden to be discarded at physics’ behest, like firing a maid when she no longer satisfies the boss’s desires.

Enlightenment and Divinity

The Buddha was enlightened but human – no scholar says he is divine. This raises problems for a certain brand of Buddhists that I have come across only in Sri Lanka. These are people who claim he knew about atomic structure, the concepts of quantum electrodynamics can be found in his philosophy and such like mumbo-jumbo. These folks don’t see the pickle they are getting into. Science is moving and changing, so if their version of Buddha is committed to current science, than in a hundred years, when science has moved on, he will be out of kilter! I wish people would think before they open their mouths.

Edwin Rodrigo wants to make me a Buddhist; he is pushing at an open door. My Marxism in no way inhibits appreciation of the Buddha’s wisdom on mind and behaviour. Phew! In one essay I have taken on Einstein, Gödel and some people’s version of the Buddha; must stop before I am certified insane and locked up in a padded cell.