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
SavagerialRepercussions 
amazing views… i hope it reached people who want merely wants to extract corporate value from mathematics
A Brilliant read! And leaving the content aside, I must say, I admire your language. Beautifully worded. No words that I cant understand, yet beautifully and artistically written. 🙂
@Ashwin: Thanks man!
Mathematics would be the language in which Nature wants to communicate to us. Not necessarily the language it communicates with itself. So I am of the view that, every molecule, every atom minding its own business, leads to this beautiful world! So in every way, Nature does not decide or compute to accelerate a ball at 9.8 m/s2. All it does is, it tells you, “Hey look, this ball is going to fall and accelerate at 9.8 m/s2.”. Would you agree with that?
Firstly, nature has been around much before mankind or life existed. Secondly, I am under the impression that mankind itself is nature in certain ways, meaning, I wouldn’t make the general distinction between natural and man-made. (Mankind is a product of natural evolution) And yes, I would agree with what you say.
Beautiful Read. Honest and Clean writing.
Here’s a different opinion.
This was a quote by Euclid that fascinated me :
“The laws of nature are but the mathematical thoughts of God”.
And this quote is something I believe in. I believe that every thing, even every little thing in the world will have a mathematical reasoning and can be *eventually* reduced to a few equations. So every aspect of nature, every design, I believe can be quantified by equations.
But by when we will learn these equations, now that’s the real challenge.
@Vasant: My view isn’t entirely different from yours. Even I believe that ultimately everything can be boiled down to equations.’Few’ being subjective, I don’t really believe in the idea of unification theories.
This takes me to challenge the entire hypothesis – Why should everything be logical? Should everything follow the rules of logic?
I do not wish to endorse the views of a philosopher, but being a science guy, I want to find out how nature works. If there are ways in which nature behaves non-mathematically, I guess we would have to accept it right.
Reblogged this on peyami.
You write very well, Vignesh 🙂
A lot of enchantment with mathematics is romantic and not mysterious at all. Mathematics is simply a well-ordered language we use to describe things we see. Well-ordered is subjective (or somewhat objective through shared subjectivity) and a language expands to fit the reality we experience. Consider English.
1) @Vignesh: We understand “blue”. Someone taught it to us by showing the sky, or a crayon saying “this is blue”. One day we see the sea and we automatically say that is blue. There is nothing intrinsically powerful in the language that crayon-blue and sea-blue are the same. Crayon and the sea have no relation but we still apply crayon-blue concept and infer that the sea is blue. It is a wonder of nature that blue is produced in different ways in the crayon and in the sea but that is beauty in nature. Not beauty in language. “pi” is just like “blue”. Its a word used to describe something we see. My view is that it occurs in a circle and also a series isn’t very surprising – sometimes we say we “feel blue”. Humans have a vocabulary and they use it to describe the reality they see. Math is obviously highly ordered and hence is less flexible than natural language. But it is the same principle. The same word might occur in different places. That is the power of math – it has well-defined words (discovered is specific circumstances but are general in reality and not tied to the circumstance of discovery).
2) Once we have basic vocabulary, we can begin to construct sentences obeying rules of language but conflicting with reality. I can say, “the leaves are blue” which is linguistically (and mathematically, in this analogy) correct but it conflicts with reality. I can construct the following two sentences a) The metals flying of fire b) The metal flew using fire. a) is linguistically nonsense (disproved conjecture – proof is merely saying that it obeys rules of the language). b) is linguistically OK but nonsense based on experiences until I invent the plane. This is the power of math, it lets you construct things from smaller things which *might* correspond to reality. Applied math is like saying “bring me the pacemaker” it is useful and produces benefits in the world. pure mathematics is like saying “but to be young was very heaven!” its beautiful but has no practical use. A language might be used to produce results but we can also play with the language itself. And we will sooner or later stumble on something which is also true. but a writer doesn’t need to also be a journalist. truth isn’t binding.
3) @Ashwin: Logic of nature is impenetrable through math. Only the logic of math is penetrable. When someone says “X was sit in the chair” you know its wrong but when someone says “X sat in the chair” as opposed to “X sat on the chair”, both of which are valid linguistically, the only way you can evaluate truth is through experiment and experience. So, nature is neither logical nor illogical. our descriptions of nature might be logical or illogical. If there is something which is true but not expressible in our language logically, we expand the language (like it happens in math) through new words or new rules. Mathematics is merely a language – albeit a very precise one. Sanskrit is a very precise natural language and many sages have sung that god thinks, speaks and sleeps in Sanskrit just like scientists swear nature speaks in math. It is merely our projection on what we study. Becoming attached to a language is common and romantic and fantastic because language reveals the beauty of the world to us and well-ordered language blurs the distinction between itself and what it describes.
4) @Vasant: Euclid was being romantic when he said that. But if he was literal, it is equivalent to saying “The laws of the world are the thoughts of god expressed in Java” because the N-body simulator was written in it. We describe in math, java, sign language etc. Nature and God are mute. So, nature is under no compulsion to be mathematically reasonable. Because math is only a description. A theory of everything is just compression of sentences. For instance, I can list “1 2 3 … 1000” or I can say “1 to 1000”. In the latter case, I have defined precisely “1”, “to” and “1000” in a way which is consistent with the rest of the language. I have achieved a compression of 1 to 1000. If I want to obtain a compression of everything is see and experience, I need a well-ordered precise language like math (+ future expansions). A theory of everything is not having the math-hammer and trying to shape the universe into a nail. It is figuring out what the universe is, and then shaping math accordingly. Logic, reasonable, mathematical etc. are all linguistic aspects. They are only a property of the language. Nature itself is perfectly free. So, definitely we will have a mathematical reasoning for everything – not because nature is obsequious to math but because we will cheat by expanding math. Just like the person who cheated by deciding he is going to call the weird feeling in him “blue”. And then marveled that the english language is powerful, and this feeling is a logical and legitimate because there is now a word for it.
And that concludes my vandalism of your blog.