Mathematics a Language of Natural Consequence
Stephen Hawking has opined “GOD plays dice”. Mathematicians see the result of throw of a dice as a probability. But, a physicist would like to analyze the problem as that of rotational dynamics considering force applied to throw the dice, location of the centre of mass, initial position of dice, and the nature of surface on which dice strikes after fall, and with these inputs he predicts or decides certainty with which result of throw of the dice. In Indian mythology Shakuni of Mahabharat perhaps knew the science of dice and that helped Duryodhan to win Chausar Game against Yudhishtir.
In real life be it any field right from universe, to biology to, social sciences to mathematics and physics every happening involves multiple inputs which are still unknown. It is easy to disowns ignorance by calling the happenings either as a destiny or an act of GOD while, aim of science is to discover the truth, the unknown relationship between various variables responsible for a happening. Scientist, verify these relationships by repetitive experiments before it is promulgated as a theory or principle. Engineers utilize these theories in various permutations and combinations to achieve desired process, action or system through inventions or improvement in existing systems. Technologists, on the other hand aim at making a process, action or system which is techno-economically viable, sustainable and reliable.
Behaviour of a system is analysed with three kinds of variables viz. a) control variables, b) disturbance variables and c) state variables. Context and the status of these state variable at a point of instance together with parameters of a system are the basic inputs. Disturbance variables are those which cause change in the state of a system and in turn influence its behaviour. Control variables are those which are used to regulate behaviour of the system in a desired manner; it virtually cancels effect of disturbance to regulate the system into a desired state. A system is nothing but a complex combination of matter and energy, which were considered exclusive till Newtonian mechanics, and Theory of relativity added a new dimension to it by propounding their convertibility. The theory takes into cognizance of the state of the system to quantify relative change in it considering only velocity of light as constant. This convertibility is universal in nature the theory if relativity gave new understanding of nature, and it helped in peeing deep into universe.
Recognition of relative difference between two variables or same variable at different point of time is not new. It is an integral part of behaviour of every living being. It is seen that every child responds to different object or situations differently. This behavioural response gets manifested in language as degree absolute, relative and superlative degree. Language, a means of communication of one’s own experience or understanding is subjective and, therefore, it lacks objectivity and we call it behavioural relativity. Psychologists have devised methods of quantifying the behavioural response through various expectations. But, the number of variables and their interdependencies makes behavioural study probabilistic. Scientific objectivity, and reproducibility of results makes physical science deterministic in nature.
In evolution of mathematics, relative difference between two similar that observations gave rise to concept of quantity which got manifested into numbers; and thus basic mathematical operations viz. add, subtract, multiply and divide evolved into arithmetic. Handling every set of specific observations, as it grows, became increasingly difficult. This was resolved by mathematicians into a generalized language algebra which facilitates expression of relationship between different variables. Algebraic expressions offer convenience of application to a specific set of data without loss of accuracy and reproducibility of results. On the other hand observations of shapes of matter seen in nature led to evolution of geometry. These three branches together led to evolution of Pythagoras Theorem, while relative proportion of two variables led to Ratio and Proportion. Increasing use of variables under different condition caused evolution of Theory of Indices. These evolutions came in due to natural relationship between numbers, variables and shapes.
Algebra is a popular branch of mathematics which helps mathematicians and scientists to convey their observations, in nature, into equations which are verifiable. Evolution of binomial theorem, logarithm, trigonometry, complex numbers, Pi (p) and Euler’s Number (e), took at different point of time to solve different problems. Surprisingly, each of definitions is complementary to other and it is thus indicative of natural consequence of one onto another. Relation of length of side of triangle to the length of other Two sides and angle between them was evolved at a much later date both in algebra and trigonometry. But, the Pythagoras Theorem, turned out to be a natural consequence of this relationship for a right angle triangle.
This analogy is now extended to Physics, wherein invention of Simple Pendulum lead to discovery of Simple Harmonic Motion (SHM). Nevertheless, investigations into trace of a particle performing circular motion at uniform speed gave broader interpretation to SHM. Further insight into all kinds of vibrations and oscillation revealed that they all are consequence of different SHMs to give rise to theory of different kinds of waves and signal processing.
Another example of Newton’s Third Law of Motion in mechanics, Le-Chatelier’s Principle on dynamics of chemical reaction and Lenz’s Law in electro-magnetism are example of action-reaction theory finding different manifestation in to handle problems of different field of nature. There are many such analogies to substantiate natural consequence of one theory on the other to render this space insufficient. This discussion leads to inference that subject matter of mathematics and science, particularly physics, and their development are closely interdependent.
It is pertinent to remember there are boundaries for any phenomenon within which it obeys postulates. These boundaries are a matter of further investigation of laws of nature which can be generalized to connects two different set of observations seamlessly.
Extension of the statement on composition of the system to universe has now become relevant. In mythology, different incarnations of GOD are shown with an aura around them. This can be related to observation of persons of different mental capabilities. A person with higher mental capability has much higher speed of analysis to determine consequence of an action, Accordingly, one takes decision or performs action, while, a person with mental ability not so high takes longer of time to understand the situation first, and then to respond. This relative difference is clearly visible in the form of glitter on the face or in the eyes of the person. It is akin to the aura of the person. This aura, based on meta-physical approach, can be compared to the energy generated by brain, while processing at abnormally high speed, and is in conformance with laws of convertibility of matter into energy. Accordingly, it is considered reasonable to advance an hypothesis that all mythological personalities were persons with an ability to think and act at a very high speed which was considered supernatural by normal person, and those were regarded as GOD. Miracles performed by such GOD were acts in conformance to laws of nature but these laws were unknown to the common man.
Getting back to understanding of Mathematics it is the fabric of all scientific discoveries. While inventions are engineering of these discoveries for making the life comfortable, thereby tending to become un-natural. Advent of technology is to conquer the nature in a manner which is sustainable, and reliable. Nowadays, there are mathematical model of all biological phenomenon which have made possible medical investigations and bio-technological inventions. Greatest, argument on the conformance of mathematical consequence is metrological prediction about depletion of ice cap, rise in sea level and numerous biological inventions. Bio-scientists and cosmologists are using mathematical model to correlate observations and determine beginning of life in this universe. Latest discovery of Higgs Boson particle is a step in forward direction. Imperfection in these predictions and inventions signifies gray areas in understanding of mathematical relationship of variables to interpret natural phenomenon. Accuracy in quantifying these variables would only help in predicting the consequences with certainty. If at any point mathematics violates natural consequence, it would call for review of all the knowledge of mathematics, need of which, as of yet, has not been felt.
Newtonian mechanics is based on observations in immediate surroundings. Sir Isaac Newton when found the contemporary knowledge of mathematics deficient in explaining rate of change or summative changes he used his observations to find the interdependencies and invented Calculus; it is a natural consequence of infinitesimal change and found an extensive application in analysis. At this stage it is relevant to invite readers to question the obvious - how can mathematics be a language? And they would find an answer that - mathematics has its own vocabulary, punctuations and grammar, a language of solving problems; and not a problem, a commonly prevalent myth. An exception of this language is that it is unambiguous and precise.
Einstein could see beyond Newtonian mechanics and suggested General Theory of Relativity which revolutionized the understanding of Mathematics and Science over a wide range from dynamics of rigid bodies to dynamics of high speed particles to explain absolute velocity of light, photo-electric effect. convertibility of matter and energy, bending of light, black holes, using his analytical power, without experimental verification. Most of it has now become a living reality.
This article is a tribute to Albert Einstein, George Bool, and galaxy of scientists and mathematicians who spent sleepless nights to question what appeared to be obvious and evolved theories, new tools and approach, to discover secrets of happening around us with the help of mathematics - a language of natural consequence. If every student takes a resolve to not to stop at learning and mastering knowledge within known boundaries, and explore his imagination to discover nature and to contribute for sustainable development tirelessly, it would be a true tribute. Taking this spirit forward at SUBODH FOUNDATION endeavour is to create conceptual clarity of established knowledge and enhances power of imagination among students. The initiative is open to all, but it emphasises upon grooming competence among unprivileged students.
Dr. Subhash Kumar Joshi
