Large Numbers and the Nature of Mathematics

The great mathematician GH Hardy once shared an anecdote of his meeting with S. Ramanujan: I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." Indeed, each number is special if we analyse it for long enough. Ramanujan and his work are regarded by many as the pinnacle of mathematical intuition. He could see patterns unravel themselves, and had also obtained a fundamental intuition for the numbers. Many of us have been exposed to this mathematical beauty occasionally throughout our study of the subject -the so called “joy of mathematics”- even though we may not feel it. For example, a repeatedly used number like one, two or zero doesn’t merely represent a concept, a definition, an object or a transformation. It is something that transcends mere formalism, and makes intuitive sense to us. Our sense of counting and recalling numbers itself is like a remarkable insight. These experiences are in fact, a type of mathematical beauty which mostly goes unnoticed. Beauty may also be seen in elegant proofs; proofs that are simple, but still insightful. For example, the proofs below provide simple solutions without even using words.

    

[The top proof shows a triangle composed of consecutive natural numbers in successive layers, which is copied and placed beside itself to form a rectangle of sides n and (n+1), while the bottom proof uses AA similarity]

While these proofs are among countless others that may give aesthetic pleasure to mathematicians, there is another aspect of the joy of mathematics that cements its unique position as both an art and a science.

Most people have a concrete understanding of what a hundred or a thousand looks like. Maybe even a million. But in the infinities of the numbers there exist numbers so large, that they are not comprehendible. An average person could attempt to wrap their head around a billion or a trillion, and maybe even relatively reason at astronomical scales. But even at that point, our ability to relate to the number simply breaks. We are in the realm of pure abstraction, and unfathomable magnitudes.

For example, consider Graham’s number- it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space(1.616255(18)×10−35 m). But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Graham’s number is one among a plethora of such large inconceivable numbers, not to mention the infinities in between.

Humans are simply dumbfounded, but also fascinated by these numbers in an almost childlike manner. Greek philosophers belonging to the school of Pythagoras believed that numbers are linked with creation, and are like the elements of the universe. People had the myth that if they knew the names of large numbers, it gave them control over the numbers and consequently, nature. As per Buddhist mythology, Gautam Budha had named a series of numbers starting with one million and reaching numbers comparable to the googol(10^100).

Contemporary mathematicians come up with concepts to explain and represent what we experience but cannot comprehend, which enunciates the scientific mindset necessary for mathematics. They are assisted by supercomputers with computational powers that were unfathomable in the past decades. Even with these technological behemoths, it becomes exceedingly clear that the more we know, the more we know that we don’t know.

This fascination with large numbers, represents one of the most accessible mathematical curiosities, in that it requires no prerequisites to understand. The other areas of mathematics are this captivating too, but require much more prior knowledge to appreciate.

Such discoveries represent human endeavour into unknown territories, constructions of entire logical systems from axioms, and also inspire us to continue to pursue them. Mathematics is a unique blend of creativity, logic, curiosity and elegance; the intersection of an art and a science. The universe is but a part of the mathematician’s playground.

References:-

(1) https://en.wikipedia.org/wiki/1729_(number)

(2) https://en.wikipedia.org/wiki/Graham%27s_number

(3) https://artofproblemsolving.com/wiki/index.php/Proofs_without_words

(4) https://math.stackexchange.com/questions/3329190/visual-intuition-for-the-sum-of-a-finite-geometric-series

(5) Brian. Clegg, A Brief History of Infinity, The Quest to Think the Unthinkable, Robinson, London, 2003.

(6) http://www.thenagain.info/webchron/india/Buddha.html, accessed 10-02-2014