An Eternal Golden Braid
Piecing together quotes, anecdotes and opinions in a juxtaposed narrative, Karthick Ramanathan tries to unravel the way we look and should look at Mathematics.
The Café Josephinum was a weekly meeting place of the ‘Vienna Circle’, a group of early 20th century scientists, mathematicians and philosophers seeking to radicalize the adoption of a scientific world view. The members considered themselves as conceptual revolutionaries with a proud and unabashed antagonism towards metaphysics. The torchbearers of a revolution seeking to force men to revolve around truth and logic, shake off centuries of misguided faith and tear apart the shackles of religion. And it was a place where a young Kurt Gödel frequented every week, with nervous excitement. He went there not to seek refuge in their philosophy, rather to hear things which transported him to his perfect Platonic world of mathematical reality. The place where he was the most comfortable. The rhythmic motion of air vibrating his eardrums and congealing in his hippocampus into something much more real than everything else in his life. The most tangible form of intangibility that he can touch; a flawless reality incapable of deception.
PLATONISM
Plato believed that all perfect mathematical structures occupied a separate idealized world filled with perfect circles, iota and the number pi. A world adorned with pure truth, proofs and logic, impervious to human tarnish. A world as precise and clear, as our physical world is nebulous at the smallest scales. But is this world real? Or useful in any manner? Platonic reality provides the blueprint according to which modern science evolves. A script for the language of Nature. It brings about an awareness of the approximations that we deal with every day. And in science, such awareness is of paramount importance.
Plato believed that this world exists independent of human thought, in a sense that thought is capable of only discovering that world’s structures; it doesn’t have a thorough conception of details of all these structures. The most remarkable and popular example attesting to this fact is the Mandelbrot Set. Benoit Mandelbrot had no prior idea that checking the region of convergence of a function z² + c, i.e starting at z = 0 and generating a set of c values for which the sequence converges, would generate an image so rich and vivid, that it can never be faithfully reproduced by human mind or even a computer. It exists only in Plato’s world, pure and complete, revealing itself only in parts to the human mind. The picture bypasses the mathematical jargon inaccessible to the layman, and registers itself through an artistic lens that we are all so comfortable in, while appreciating things. The simple mathematics that leads to this is no less beautiful.
FORMALISM, ABSTRACTION AND RIGOR
Srinivasa Ramanujan never had any proper mathematical training in his childhood. Ideas just emerged to him whole, without justification. Completely isolated from the mathematics of the West, more often than not, he ended up rediscovering already proven results. Ignorant of even in the most ubiquitous mathematical terminologies we study in our secondary school, he had to develop his own mathematical machinery to pen down his findings. His first contact with a western mathematical text was a rather mundane undergrad level text called Synopsis of Pure Mathematics (1886) written by G.S Carr. The book was a compilation of fundamental theorems in mathematics that had been developed till that point of time, but without any proofs. To any other student, the book was as boring as any cliched math text could get. To Ramanujan, it was gold. He sat down and re-derived almost 300 years of mathematical results in a few months as a teenager.
As Grant Sanderson (creator of the math YT channel 3Blue1Brown) puts it,
“When you rediscover math on your own, when you next encounter them, you will see them as familiar friends, not as arbitrary definitions. So be playful!”
For anyone who hasn’t seen his videos, they are — and I wish to put this no less dramatically — pure catharsis.
Mathematical jargon is just machinery to make arguments airtight and unambiguous. To the extent to which the field has developed till now, it is easy to get caught up in formal language and lose the theory’s underlying meaning. Everyone unanimously can attest to the trepidation with which a university freshman views the epsilon-delta definition of continuity in an introductory calculus course. Why are such ‘basic’ notions written down in such painstakingly laborious ways? Mathematics must be immune to the inadequacies and ambiguities that plague our language. This is where formalism and rigor come in. They make a system airtight within the axioms it operates within. Anyone must be able to objectively gauge the truth or falsity of any structure. Anyone who digs in the same spot in the same manner must discover the same things. Our intuitive compass often fails to guide us as we venture deeper into the forest of a mathematical field. Even intuitively trivial notions might not always be as simple as we see them to be. So, we need to develop a machinery which would work in any part, though it may be frustratingly slow to work with in the beginning of the forest where is hardly any need for it. A seemingly intuitive extension of the addition law for vectors in an infinite dimensional space is not permitted within the axioms of vector calculus (which particularly stump freshmen in its abstraction). If you find this perplexing, try playing around with alternating harmonic series to discover the subtleties involved in infinite summations.
Newton had to develop calculus as there were no mathematical tools available for him at that time to quantify his physical ideas. Math has always grown more and more abstract since then, progressing for its own sake rather than because it is required for some physical theory. And in the process, it has grown more and more powerful. As one of our favorite professors here puts it,
“Mathematics is far too powerful for our good. It gives us results, without us really understanding what we are getting.”
It is extremely important for us to periodically step back and get an intuitive understanding of what we are trying to do, while dealing with abstract mathematics. It is a pernicious beast during application. Why is π, a number associated with circles occurring in every nook and corner of physics and mathematics. If we look deep enough, there is a circle embedded in each and every one of those places. Quantum mechanical calculations are easy enough to follow for any person with knowledge of linear algebra. What is important is interpreting the meaning behind the results which almost seem to be magical. Funnily enough, Morpheus captures this on some level when he says,
“Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.”
When you just start believing in mathematical results without any proper understanding, you just are following a religion. Much of math we learn now like linear algebra and complex analysis are extremely fundamental and deep. There is only so much that can be taught in a class and it is a worthwhile endeavor to dig deeper to derive the true essence of these topics.
INCOMPLETENESS
Gödel would have been perspiring as he grappled with the Liar’s Paradox amidst the Vienna Circle, whose members are deeply skeptical and scornful of such things. Is the person who says ‘I am lying’, a liar? Such paradoxes arising out of self-referential statements are just borne out of the language inadequacies. Mathematics is complete, it has no place for them. Nevertheless, more sophisticated questions such as the Russell’s Paradox — ‘Does the set of all sets which contain itself, contain itself?’ began to leave mathematicians squirming. Hilbert attempted to clarify this foundational crisis in mathematics by posing a set of questions which sought to address completeness, consistency and decidability in mathematics.
Gödel managed to prove mathematically that there exist true statements within a system which can never be proven. They exist forever out of the reach of mathematics. Consider the statement, “This statement is non-provable.” If it is provable, then we are proving a falsehood, which is extremely unpleasant and is generally assumed to be impossible. The only alternative left is that this statement is unprovable. Therefore, it is in fact both true and unprovable. Our system of reasoning is incomplete, because some truths are unprovable. This along with Turing’s result on the undecidability of the Halting problem buried a mathematician’s notion of pristine logic.
Gödel was a convinced theist. Our minds can see truth. See it even when mathematics cannot. Our minds are bigger than that. Perhaps, math is sufficient only unto math.
“On one stave, for a small instrument, the man writes a whole world of the deepest thoughts and most powerful feelings. If I imagined that I could have created, even conceived the piece, I am quite certain that the excess of excitement and earth-shattering experience would have driven me out of my mind.” — Johannes Brahms on Bach’s Chaconne in Partita №2 in D Minor.
BUT IS ALL THIS REALLY USEFUL?
In September 2019, a ‘huge problem’ was solved in number theory by a team led by Andrew Booker from the University of Bristol — the solution set (x,y,z) to the equation x³ + y³ +z³ = 42. The solution for every number between 1–100 had been found . Only 42 remained. After months of searching in large supercomputing facilities, the solution x = -80538738812075974, y = 80435758145817515 and z = 12602123297335631 was found. This is the world of pure math. Is doing all this worthwhile in any manner? I really don’t know! Maybe I can persuade you to read G.H Hardy’s essay ‘A Mathematician’s Apology’. Sometimes while obsessing over practicality, we hardly leave any room for a serendipitous discovery of an interesting idea.
Take the example of another story from last year, in which things happened the other way round — physicists investigating neutrino physics ended up discovering an unexpected relationship in linear algebra, something that which mathematicians have been dealing with for centuries.
While thinking upon some fairly random problem, we might unknowingly stumble into something quite deep. An important piece of math is usually connected to several fields and is quite fundamental. Considered most important of these now, are the 7 ‘Millennium Problems’, for each of whose solutions, the Clay Mathematical Institute offers a million dollars as reward. Only one of them has been solved so far — the Poincaré conjecture, by the mysterious anti-social Russian mathematician Grigori Perelman, who refused both the million dollars and the Fields Medal, having no interest in them.
Terence Tao, the prodigious Australian mathematician once remarked,
“Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories and anecdotes are important to the young in understanding real life.”
If a ‘useless’ problem can inspire you to think deep, who cares if it is useless. Curious souls can check out Terry Tao’s blog post on the ‘neutrino’ discovery.
Math is everywhere. It is behind the scurrying of the cockroaches in Perelman’s bug infested apartment, behind the construction of the gopuram of the Sarangapani Vishnu temple where Ramanujan used to scribble his equations, behind the poison consumed by Turing to end his life and the obsessive fear of which ended Gödel’s. Behind the workings of our minds and hearts too.
Maybe this is a fable as well. And if it can inspire you to even develop a myopic version of Gödelian eyes, looking at math with unabashed fidelity, who cares if it is written by only a paltry undergrad with only limited ability but unlimited curiosity?
This article is inspired by several works, most notably by the book ‘A Madman Dreams of Turing Machines’, by Janna Levin. Do give it a read!
