By 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 a few others have been less cruel firing only a few volleys. There have been contributions in web Comments (Siri Gamage, Lester, Dr M. Gonlaskorale, AVB, and Edwin Rodrigo to name a few). I declare upfront that I cannot do justice to all for two reasons. First, I am not conversant with spiritually oriented topics such as the “Stages of Understanding in Buddhism” or “alternative non-Western knowledge systems” (sorry Edwin and Lester) and am a layman on the brain’s “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 elsewhere.
Secondly, within the limits of my column, I have to be restrained and not bite off more than I can chew. It is also not possible within reasonable length to explore minds like that of Alan Turing, inventor of the computer and founder of systematic computing (see Alan Turning: The Enigma, by Andrew Hodges, Princeton University Press) or discuss 21-st Century artificial intelligence (AI).
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 already contained in the scientific theory itself? I say NO it cannot, 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 maths 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 this contrarian chap and would to go further and replace ‘purely scholastic’ with ‘utterly meaningless’. Regarding the Einstein quote above, unless it is a fabrication it 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 (λ), which later he shamefacedly admitted was “my greatest blunder”. David Hilbert, the leading mathematician of the day, derived the general relativity equations at the same time but separately from Einstein but did not make this blunder.
The use of maths, statistics, computing and AI to implement the marvels of modern technology belongs to a separate genre different from the scope of this essay.
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) asks and is agnostic. 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”.
Left to right: David Hilbert, Alan Turning, Kurt Gödel, Bertrand Russell
ML confuses himself. Are Maxwell’s Equations (ME) physics or mathematics? Their beauty can banish Cleopatra from any bridal chamber but must not seduce us to forget that the body within is physics, the raiment without radiant mathematics and can be written in two forms, differential or integral.
Forgive me this one unavoidable paragraphs; I will then get back to terra firma. The first equation says “There is no true magnetism”; meaning magnetism is a product of electric currents. The second equation says the electric field emanating from any region depends on the net charge 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 fourth equation supplements the first one and says how much magnetism is made by a given electric current. There is no need for readers to follow the mathematics 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 donned 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 but this would be too long-winded a digression to undertake here).
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.
- If a system is consistent, it cannot be complete.
- The consistency of its axioms cannot be proved within a system
The concepts are self-evident. Obviously the axioms of Euclid’s geometry cannot be proved within Euclid’s geometry! (Gödel’s terrain was number theory). What is the relevance of all this to our science-maths concerns? If a scientific revolution causes one maths to be thrown out and another to find favour, the former’s axioms go out of the window as well. Euclid’s axioms lost out when general relativity lent on Riemann. Unexpectedly, some branches of abstract mathematics earned handsome dividends when quantum physics blossomed; most others remain abstractions solely for the delight of mathematicians. Maths is a handmaiden to be discarded at the behest of physics, 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 encountered only in Sri Lanka. These people claim that he knew about atomic structure, that the concepts of quantum electrodynamics are contained in his philosophy and such mumbo-jumbo. These folks don’t see the pickle they are getting into. Science is 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 is no way hinders an appreciation of the wisdom of the Buddha. 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.