On `Mathematics'

Department of Physics & Astrophysics

University of Delhi, Delhi - 110007

E-mail: [email protected]

With Manjul Bhargava winning the prestigious Fields Medal, Subhash Khot bagging the Rolf Nevanlinna Prize and Ashoke Sen receiving the Dirac Medal, mathematics has become temporarily a source of non-trivial excitement among the young, in the land of Srinivasa Ramanujan. Ashoke Sen, an internationally renowned string theorist, is not technically a mathematician. But that is a matter of little consequence as string theory, though a branch of physics, is almost inseparable from subsets of advanced pure mathematics. To gauge its importance to mathematics, one may recollect that Edward Witten, an eminent string theorist as well as a celebrated theoretical physicist, was awarded Fields Medal in 1990 for his seminal papers on supersymmetry, Morse and Hodge-de Rham theories that led to progress in number theory and complex analysis.

Why is it that a fraction of gifted people, however miniscule, keep scaling newer heights of an esoteric and difficult subject called mathematics? A George Mallory type answer `because it is there’ notwithstanding, one of the reasons could be the inherent affinity in many for tackling riddles, whodunnits and paradoxes. Some may recall cracking their heads once upon a time on conundrums associated with unexpected-hanging puzzle or Klein bottles from Late Martin Gardner’s feature `Mathematical Games’ (which, later, made a metamorphosis to `Metamagical Themas’, an anagram of the original title, in the deft hands of Douglas Hofstadter, after he took charge of the column) that appeared in every issue of the Scientific American magazine from mid-50s to 80s. This column initiated several young students to paradoxes like Bertrand Russell’s alluring poser involving a set of barbers who shave only those men who do not shave themselves. Should a barber, belonging to this set, shave himself? Be prepared to tie yourself into knots fathoming this one! Russell’s paradox had serious implications in axiomatic set theory.

Returning to the nature of mathematics, evidently arithmetic and geometry grew out of necessity. Early humans not only had to keep track of their possessions through counting but also needed to estimate directions, distances, shapes and sizes, without which hunting, exploration, building cities like that of Harappa and Mohenjo daro, raising humongous Egyptian pyramids, etc, would not have been possible. Homo sapiens who could assess numbers and sizes, and discern shapes and directions, had an evolutionary advantage for survival during the Darwinian struggle for existence.

But as is the wont of human brain, brighter of the lot, dealing with this incipient subject of numbers and shapes, perceived interesting patterns in some of these entities and their interrelations, that entailed concepts such as the zero (an Indian gift), prime numbers, the Pythagorean relation between sides and the hypotenuse of right triangles, and so on.

Playing and tinkering are natural human instincts and so, not surprisingly, curious and innovative minds toyed around with patterns found among the arithmetical and geometrical entities to create rich logical systems that could establish non-trivial results (i.e. theorems) such as the number of prime numbers being infinite or the Pythagoras theorem, by deploying imaginative and clever tricks on a set of very few, almost self-evident assumptions (i.e. axioms).

The Midas touch of mathematical minds added fillips to axiomatic systems, like for example Euclid’s geometry, to grow wings as though of their own and fly out to magical and intangible worlds. Numbers and geometrical concepts got transmuted and generalized to abstract but beautiful creatures, seemingly far removed from concrete reality. To cite a case, the measure of distance between any two points that relied on Pythagoras theorem in the standard Euclidean geometry got generalized to abstract ones involving metric tensors suitable for curved and warped spaces described by non-Euclidean geometry. The genie of mathematics was out of the bottle (Klein’s?) !

Were human beings genetically programmed to be abstract mathematicians or theoretical physicists? Instead, suppose we ask: were we genetically wired to have been swayed emotionally by sophisticated music? Putting forward a thesis that music has its roots in the sequence of notes present in bird songs, and that those early humans who were sensitive to and were drawn to simple melody of a koel's cooing or of other singing birds, had greater chances of survival, finding mates as well as passing on their genes to offspring (since birds gathered in regions where water and food are abundant), can explain why music affects us emotionally and why its primitives are similar to bird songs.

With time, simple tunes grew in richness because of the fascinating flexibility of brain which grows more neuronal connections, with extra stimuli provided by the inputs and outputs from other musically minded people, leading to further creative activities. The ever increasing complex networks both of inter-neuronal highways in a brain as well as of musicians resulted in increasingly sophisticated body of work in music. Contemporary music is obviously far more complex and richer in comparison to the brief melody of a bird song. It will not be far fetched to expound a similar theory in the case of mathematics for its ever growing complexity. So, what started with an evolutionary advantage, became richer, more abstract and multi-layered over time.

Modern mathematics is not a recondite recreation akin to the game of chess (yet another Indian innovation). It is a fact that all fundamental laws of Nature are expressed in mathematical terms. As pointed out emphatically by the eminent physicist Eugene Wigner in a lecture in 1959 about `The unreasonable effectiveness of mathematics in the Natural sciences’ , that abstract concepts and relations created in contemporary mathematics for its own sake, turn out at times to describe fundamental truth concerning aspects of the real universe. Although, it should be remarked that only a miniscule portion of the vast body of mathematical creations get to enjoy this status. Therefore, one can go ahead and claim that it ought not to perplex us that a small subset of mathematical ideas based on beauty and abstract generalization of concepts rooted to reality, get realized in fundamental physics.

It so happens that physical entities are measured quantitatively (i.e. in terms of numbers), and hence their interrelations, including temporal cause and effect links, better be based on a language that is numeric, precise, logical and unambiguous. Clearly, mathematics is best suited for this. One wonders about the effect of Godel's theorem (which, simply stated, implies that in a consistent logical system there will exist well formed expressions that can neither be proved nor disproved) on physical theories. So, does ambiguity in the physical world enter through a Godelian back door? Or, is it that whenever an undecidable statement springs up in a physical theory, one simply subjects it to an experimental test in order to obtain its truth value? For, Natural science has the luxury of experimentation!

The real world has always continued to inspire mathematicians, whether one considers the birth of calculus for finding trajectories of bodies moving in gravitational field or of distribution theory that ensued from Dirac delta function. It is common wisdom that systematic analysis of gambling outcomes by Fermat, Pascal and Huygens had ushered in the mathematical theory of probability. In a lighter vein, the Mahabharata hero Yudhisthira could have outwitted Shakuni in the game of dice had he brushed up on the theory of chances.

One wonders whether the chance coincidence of angular size of sun and moon being the same at present, causing eclipses to occur, played a significant role in the development of mathematics. After all, mathematicians of repute exercised their minds to understand and predict eclipses, be it Hipparchus, Aryabhatta or Varahamihira. In other words, it is scientifically relevant to ask whether in the absence of moon occulting the sun, there would have been enough impetus and motivation to develop trigonometry and other useful computational techniques. We will probably never find out.

Descartes had amalgamated algebra and geometry to create coordinate geometry. On a lighter note, combining the utterances of Descartes and Archimedes, one may envisage a maxim `Cogito Ergo Eureka’ for the pursuit of mathematics!