Mathematics is The Order that governs the Universe. Can it be a Framework for Experience?
M.R. Subramanian aka Ramanathan S Manavasi
The Thought Provoking Article of Rajesh :
Rajesh Kasturirangan in his fascinating post “The Tyranny of experience” mentions “There’s no greater tyranny in my world than that of experience, by which I mean the ever more sophisticated versions of the claim “seeing is believing.”
https://rajesh.io/the-tyranny-of-experience-8e2c6b0b2699)
“We are skeptical by nature, demanding evidence and proof and subjecting claims to all kinds of tests before we believe them. Experience produces facts, in principle if not in practice. The power of experience lies in that it generates both doubt (I need to see in order to believe) and faith (once you see you have to believe). I will come to the structural blindness of experience seeking in a moment; let’s first ask what lies beyond experience. Mathematics. Numbers are not experienced and yet some of us believe in their existence. How’s that possible? Experience only reveals objects with shape and form but numbers have neither. What form of existence can a shapeless formless entity have and how is such an entity revealed to us? Then there’s the fortunate circumstance that math seems to work. It helps us design satellites and nuclear bombs and other nice things that help us rule the earth and stalk the heavens. We can not deny math in the way we can deny god(s), but mathematical entities are not experienceable. Or are they?”
All of Rajesh’s articles in Medium Website are thought provoking and can be used for brainstorming. They are meant as a bed rock for further thoughts, analysis and introspection. I started my voyage of introspection to find out the truth about the role of mathematics in shaping the various patterns leading to intelligence. I intend dwelling on the achievements of Srinivasa Ramanujan whose intelligence was realized through deep intuition or divine inspiration. I hope to explain the various layers of meaning in this regard, one layer at a time. My main purpose is to illuminate the audience for this responsibility.
Spiders. River, Waterfalls — The underlying mathematical patterns :
There are over 40,000 species of spiders and each spider can produce up to seven different kinds of silk. Thus, more than 99.99 percent of spider silks are as yet unexplored. Scientists have found the toughest material made by life yet — the silk of a spider whose giant webs span rivers, streams and even lakes.
Spider silks were already the toughest known biomaterials, able to absorb massive amounts of energy before breaking. However, researchers have now revealed the Darwin’s bark spider (Caerostris darwini) has the toughest silk ever seen — more than twice as tough as any previously described silk, and more than 10 times stronger than Kevlar.
Think of a spider spinning a web and ask yourself, Does the spider have knowledge about Hooke’s Law regarding the tension of stretched strings ? The spider must ‘know’ about this in some sense, otherwise the web would fall apart. Would it be more accurate to say that the spider’s brain has tacit, rather than explicit, knowledge of Hooke’s Law. The spider’s brain (yes, it has one) has no explicit representation of it. It can only weave webs according to a fixed motor sequence. The human’s deployment of Hook’s law is open ended and flexible, available for an unlimited number of applications. Most of the world that we have falls in between these two extremes : the mindless knowledge of the spirit and the abstract knowledge of the physicist. We are limited by our sensory abilities, by our species membership, by our narrow attention — at least the last of which can be overcome.
We have long assumed that, like many invertebrates, the spiders are little more than automata, lacking an inner life. These creatures possess an extraordinary kind of consciousness, including minds that extend beyond their bodies. We are now discovering that some arachnids possess hidden cognitive abilities rivaling those of mammals and birds, including foresight and planning, complex learning and even the capacity to be surprised. The delicate silk threads they spin out behind them, so easily swept up by a feather duster, help them to sense and remember their world. The surprising fact is that the spiders’ silk is so important to their cognitive abilities that some scientists believe it should be considered part of their mind. We must see the spiders as one of the most ubiquitous, important and vilified groups of animals that have ever evolved. What is more, these incredible creatures could also challenge our understanding of our own intelligence and minds.
There is a mathematical order inherent in our universe. Let us start with rivers. If we measure the length of a river and divide by the direct route from the start to the end we’ll get its sinuosity or bendiness. Mathematicians found that the average sinuosity of every single river in the world is pi. The same mathematical constant used to calculate the mass of an electron and the gentle breathing of a baby helps define how bendy all rivers are. They represent the inherent order of so many things in nature.
Not all water is wet. It is this specific arrangement of molecules caused by surface tension that gives it the emergent property of wetness. If we were to arrange water droplets together in a certain way we could get a waterfall. All waterfalls across the world fall at the same speed because they are all subject to the same acceleration, gravity. The Navier-stokes equation describes how all fluids like the water in a waterfall will move and behave. It also explains everything from how blood flows in our bodies to how we can best simulate water in video games.
Mathematics is just the language that helps us interpret all this activity. So the question is can math also help us understand how our own brains work? Can we quantify the feeling of meeting someone special for the first time or the feelings we get when we spend time with them? Consciousness is a mathematical pattern. When information is being processed governed by a set of patterns like those found in the branches of a tree or the fluid dynamics of a waterfall the way it feels is consciousness.
It is a beautiful emergent phenomenon arising only when molecules are arranged in a certain pattern just like how wetness arises only when water molecules are arranged in a certain pattern. Math gives us a framework to reason about what the order behind these patterns could be and we can use the beauty that’s all around us as our torch in this journey. One that we hold up and follow with the belief that eventually, it might just lead us to the truth.
Srinivasa Ramanujan — an extraordinary harmonious blending of Reason and Revelation :
What is the source of scientific inspiration? Are we the mental recipients of a reality out there, which is communicated to us through the unfolding of consciousness, whether in the form of deep intuition or divine inspiration? Does discovery arise out of knowledge, or does the unknown define the nature of discovery? Is the brain a receiver, or a channel of truths arising out of a dimension beyond the physical? How does the individual brain interpret and represent new information from beyond the frontiers of the known?
The great mathematician Ramanujan’s name is remembered everywhere around the world, (even if some might disagree) for these things :
•Magic Square
•Brocard — Ramanujan Diophantine equation
•Dougall — Ramanujan identity
•Hardy — Ramanujan number
•Landau — Ramanujan constant
•Ramanujan’s congruences
•Ramanujan — Nagell equation
•Ramanujan — Petersson conjecture
•Ramanujan — Skolem’s theorem
•Ramanujan — Soldner constant
•Ramanujan summation
•Ramanujan theta function
•Ramanujan graph
•Ramanujan’s tau function
•Ramanujan’s ternary quadratic form
•Ramanujan’s prime
•Ramanujan’s constant
•Ramanujan’s sum
•Rogers — Ramanujan’s identity
Srinivasa Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years. His leaps of intuition confound mathematicians even today, a century after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas- polymer chemistry, computers, astrophysics, molecular physics, even (it has been recently suggested) cancer — scarcely imaginable during his lifetime.
Once G. H. Hardy rated his contemporary mathematicians based on pure talent. Hardy rated himself a score of 25 out of 100, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100 ! Hardy also said that Ramanujan’s solutions were “arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account”
S.Chandrasekhar, Indian Astrophysicist, Nobel laureate 1983, told thus: “I think it is fair to say that almost all the mathematicians who reached distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example. Even those who do not know about Ramanujan’s work are bound to be fascinated by his life.”
The words of Hardy himself speak volumes of Ramanujan: “I have to form myself, as I have never really formed before and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another and about whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”
The life of Ramanujan is actually a textbook from which many things could be conceived. Despite the hardship faced by Ramanujan, he rose to such a scientific standing and reputation no Indian has ever enjoyed. It should be enough for youngsters to comprehend that if we can work hard with indomitable determination, sheer perseverance and sincere commitment, we too can perhaps soar the way like Srinivasa Ramanujan.
When you have a room with a number of superbly drawn paintings , each perfect in its own regard, you can enjoy it and you know everything has its own value and each being diverse in what it offers, has a unique contribution towards the beauty and perfection of that room. The moment we remove even one of those paintings out, let it be the most simple or the most complex, we are in a way disrupting the rhythm, and the room is no more perfect, there is a vacuum, and for someone who has enjoyed the beauty of that room immensely, no other painting can give the room back its original perfection. Likewise, history especially the history of mathematics has unfolded in such a way that all those great ones from Pythagoras have been uniquely contributing to make this science beautiful. So, we cannot take even out of them, rather we can just accept the beauty of their companionship in developing Mathematics as our favourite science.
Anyway I would like to point one more thing. How Ramanujan is indispensable from that list. Well, it is simply because we see an extraordinary harmonious blending of Reason and Revelation in his works, two seemingly contradicting factors, Intelligence and Intuition! That makes him all the more Unique.
“It’s fascinating to me to explore his writings and imagine how his brain may have worked. It’s like being a mathematical anthropologist,” said Emory University mathematician Ken Ono.
Can Ramanujan’s mathematical magic be reduced, as world academia has claimed, to a ‘deep intuition’? What is intuition anyway? Could it be that the brain is an organ of consciousness and that when it’s trained to surrender, or to open in scientific “prayer”, it’s able to reflect intelligence from sources way beyond itself? Where do scientists go when they look for a flash of inspiration? To the past? To the known? Or to the great unknown, over the edges of the borders of their own conceptual frameworks?
Vision is an enormously complicated process involving at least a dozen parts of the brain, each of which recognizes something; one recognizes a horizontal line, one recognizes perspective, one recognizes color, one recognizes motion. Somehow they are all integrated together. You cannot really visualize four dimensions or five dimensions but you get a rough idea of what it is like by comparison with three.
It is possible that a single cosmic field is expressing itself as all material form and all experience of that form. It is vibrationally transforming itself into the localized field excitations characteristic of the interwoven orchestration of living-cell biochemistry while also having a nonlocal awareness of itself. This fundamental field, intrinsic to the Universe, has both an objective aspect, known as the vacuum energy /zero-point field and a subjective aspect, known as our conscious awareness.
The most honest version of mathematics is the one that grants us the freedom to revise its truths. We are bound, of course to logic — there is no virtue in refuting the irrefutable. But all of us who present mathematical ideas have a responsibility to illuminate our audience. We can start by not making mathematics seem so inevitable. In alliterative terms: “make the irrefutable illuminating by making it less inevitable.”
M.R. Subramanian aka Ramanathan S Manavasi
The author of the book “The Art of Seeing — Essence of Vision and Epiphanies of Perception”