This is one thing that has been troubling me for a very long time. What is the ratio of circumference to diameter of a circle doing when I am simply trying to find the sum of a series? What is Fibonacci’s sequence doing inside a flower? What is the golden ratio doing in my face? And these concepts did not grow out of just observation of natural phenomena. And what about the laws of nature? One cannot help wonder what Einstein once did,
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?
Whenever I think about the nature of mathematics, it leaves me with an eerie feeling. Is math just a representation of reality or is it something more natural, something whose presence is more inherently woven into the fabric of nature? Maybe the laws of nature are written in the language of mathematics to be an object of mathematics itself!
Looking at the evolution of mathematics, it once started as a means to formalize counting. Prehistoric people recognized to count physical objects and along with it, more abstract quantities like time. ‘Addition’ developed as a process of adding numbers. But we no longer retain the same view – what were once processes have now become concepts. And then on, it was an ever increasing series of abstractions, an expansion of subject matter increasing its applications from elementary arithmetic to astronomy and boolean algebra and now we are beyond applications. What once started as a tool to solve everyday issues, has now become a theory on its own, and applications are still trying to catch up with the advancement in the theory. But is it necessary for applications to catch up with mathematics? So lost are we in our mathematics, and such is the great power of mathematics for we can work with the bald abstractions, doing away with the physical reality, that we end up having to question every stage of our logical reasoning for correspondence to reality. Ironically, this is also one of the major weaknesses of mathematics. Is the role of mathematics an unadorned representation of the laws of everyday life, and that of nature and its reality? After all, of what use is something that cannot be applied to solve problems?
More often than not, scientists are more than mere problems solvers. They are truth seekers. One thing is to understand a tree; it requires more to understand a forest. Probably the same thing differentiates a specialist or an artisan from a truth seeker. Today, mathematical theory itself is advanced because and due to the requirement in its applications. Newton was credited (at least partly, though not without criticism over and with Leibniz) for the development of calculus. The invention of the path integral formulation by Feynman is another example. Subjects like operations research, statistics and computer science sprouted and have become one with certain areas of mathematics. So where does that leave us with the idea of mathematics being inherent in nature?
As a self-altercation to my mentions of Fibonacci or pi, I argue saying that these concepts are rather too simple and elementary to be of surprise to turn up in nature or any formalism for that matter. However, I later realized that there is much more than elementary sequences or ratios. Music theory of octaves, a basic miracle of music can be explained in a logical system of mathematics. The origin of quantum mechanics itself is credited to the discovery of Max Born that Heisenberg’s computation rules matched those of matrices established decades ago. The basic concepts of quantum mechanics, ‘states’ are vectors in Hilbert space which, though is a generalization of the Euclidean space is no coincidence in its frequent and natural appearances as function spaces in various theories.
We humans are ourselves innate mathematicians. Most of us can subconsciously perform complicated mechanics to calculate the trajectory of a ball to position ourselves for the catch. Or take that of a scaffolder who can perfectly understand the 3-4-5 of a triangle and know not a scratch about Pythagoras. The ancient Egyptians and Greeks recognized the golden ratio and used it in the design of great monuments like the Pyramid and the Colosseum. Why is it that our brains find something like the golden ratio, aesthetically pleasing; why is it that the octave is a perfect interval? Maybe mathematics cannot explain that right now. But lots of research in theoretical neuroscience are leading to better understanding of the brain.
The aestheticism and mysticism that surrounds mathematics start from the days of (self-conjured god) Pythagoras and it is not absolutely refutable for that stature. One cannot completely fail to appreciate the beauty of Euler’s identity: five fundamental mathematical constants, three basic arithmetic operations all appearing once, beautifully combining to give one jewel of an equation. One can only deliquesce on reading the words of Bertrand Russell:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
We have always attributed everything of nature to have a sublime beauty in it and I give in to the temptation of calling mathematics, with all its profundity and beauty, as an innate nature of Nature itself. Maybe I am biased with my belief that humans do not deserve all the credit for mathematics and it does certainly require a touch of Nature’s magic to be so divine in its acumen. I hope there is more evidence in future to point to the idea of mathematics being a product of nature and that some day someone will find mathematics embedded in the fabric of nature. At least until someone proves otherwise, I will remain in my revelry and celebrate the omniscient, alleviating and blissful eternity of mathematics.
-SR
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