I am reproducing an article, “The Universal Appeal of Mathematics — Geetha S. Rao” from “The Mathematics Student” , volume 83, Numbers 1 to 4, (2014), 01-04.
The purpose is just to share this beautiful article with the wider student community and math enthusiasts.
The Universal Appeal of Mathematics: Geetha S. Rao:
Mathematics is the Queen of all Sciences, the King of all Arts and the Master of all that is being surveyed. Such is the immaculate and immense potential of the all-pervasive, fascinating subject, that it transcends all geographical barriers, territorial domains and racial prejudices.
The four pillars that support the growth, development, flowering and fruition of this ever green subject are analytic thinking, logical reasoning, critical reviewing and decision thinking.
Every situation in real life can be modelled and simulated in mathematical language. So much so, every human must be empowered with at least a smattering of mathematical knowledge. Indeed, the field of Artificial Intelligence is one where these concepts are implemented and imparted to the digital computers of today.
From times immemorial, people know how to count and could trade using the barter system. Those who could join primary schools learnt the fundamental arithmetic and algebraic rules. Upon entry into high school and higher secondary classes, the acquaintance with the various branches of this exciting subject commences. It is at this point that effective communication skills of the teacher impact the comprehension and conceptual understanding of the students.
Unfortunately, if the teacher is unsure of the methods and rules involved, then begins a dislike of the subject by the students being taught. To prevent a carcinogenic spread of the dislike, the teacher ought to be suitably oriented and know precisely how to captivate the imagination of the students. If this is the case, the students enjoy learning process and even start loving the subject, making them eagely await Mathematics classes, with bated breath!
Acquiring necessary knowledge of algebraic operations, permutations and combinations, rudiments of probabilistic methods, persuasive ideas from differential and integral calculus and modern set theory will strengthen the bonds of mathematical wisdom.
From that stage, when one enters the portals of university education, general or technical, the opportunity to expand one’s horizon of mathematical initiation is stupendous. Besides, the effective use of Mathematics in Aeronautical, Agricultural, Biological, Chemical, Geographical and Physical Sciences, Engineering, Medicine, Meteorology, Robotics, Social Sciences and other branches of knowledge is indeed mind boggling.
Armed with this mathematical arsenal, the choice of a suitable career becomes very diverse. No two humans need to see eye to eye as far as such a choice is concerned, as the variety is staggering! So, it is crystal clear that studying Mathematics,at every level, is not only meaningful and worthwhile but absolutely essential.
A natural mathematical genius like Srinivasa Ramanujan was and continues to be an enigma and a Swayambhu, who could dream of extraordinary mathematical formulae, without any formal training.
A formally trained mathematician is capable of achieving laudable goals and imminent success in everything that he chooses to learn and if possible, discover for himself, the eternal truths of mathematics, provided he pursues the subject with imagination, passion, vigour and zeal.
Nothing can be so overwhelming as a long standing problem affording a unique solution, bu the creation of new tools, providing immense pleasure, a sense of reward and tremendous excitement in the voyage of discovery.
These flights of imagination and intuition form the core of research activities. With the advent of the computers, numerical algorithms gained in currency and greater precision, enabling the mathematical techniques to grow by leaps and bounds!
Until the enumeration of the Uncertainty Principle by Werner Heisenberg, in 1932, mathematics meant definite rules of certainty. One may venture to say that this is the origin of Fuzziness. Lotfi Zadeh wrote a seminal paper, entitled Fuzzy sets, Information and Control,
8, 1965, 328-353. He must be considered a remarkable pioneer who invented the subject of Fuzzy mathematics, which is the amalgam of mathematical rules and methods of probability put together to define domains of fuzziness.
Fuzzy means frayed, fluffy, blurred or indistinct. On a cold wintry day, haziness is seen around at dawn, and a person or an object at a distance, viewed through the mist, will appear hazy. This is a visual representation of fuzziness. The input variables in a fuzzy control systems are mapped into sets of membership functions known as fuzzy sets. The process of converting a crisp input value to a fuzzy value is called fuzzification.
A control system may also have various types of switches or on-off inputs along with its analog inputs, and such switch inputs will have a truth value equal to either 0 or 1.
Given mappings of input variables into membership functions and truth values, the micro controller makes decisions concerning what action should be taken, based on a set of rules. Fuzzy concepts are those that cannot be expressed as true or false, but rather as partially true!
Fuzzy logic is involved in approximating rather than precisely determining the value. Traditional control systems are based on mathematical models in which one or more differential equations that define the system’s response to the inputs will be used. In many cases, the mathematical model of the control process may not exist, or may be too expensive, in terms of computer processing power and memory, and a system based on empirical rules may be more effective.
Furthermore, fuzzy logic is more suited to low cost implementation based on inexpensive sensors, low resolution analog-to-digital converters and 4-bit or 8 bit microcontroller chips. Such systems can be easily upgraded by adding new rules/novel features to improve performance. In many cases, fuzzy control can be used to enhance the power of existing systems by adding an extra layer of intelligence to the current control system. In practice, there are several different ways to define a rule, but the most simple one employed is the max-min inference method, in which the output membership function is given the truth value generated by the underlying premise. It is important to note that rules involved in hardware are parallel, while in software they are sequential.
In 1985, interest in fuzzy systems was sparked by the Hitachi company in Japan, whose experts demonstrated the superiority of fuzzy control systems for trains. These ideas were quickly adopted and fuzzy systems were used to control accelerating, braking, and stoppage of electric trains, which led to the historic introduction, in 1987, of the bullet train, with a speed of 200 miles per hour, between Tokyo and Sendai.
During an international conference of fuzzy researchers in Tokyo, in 1987, T. Yamakawa explained the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an inverted pendulum experiment. The Japanese soon became infatuated with fuzzy systems and implemented these methods in a wide range of astonishing commercial and industrial applications.
In 1988, the vacuum cleaners of Matsushita used micro controllers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. The Hitachi washing machines used fuzzy controllers to load-weight, fabric-mix and dirt sensors and automatically set the wash cycle for the optimum use of power, water and detergent.
The renowned Canon camera company developed an auto-focusing camera that used a charge coupled device to measure the clarity of the image in six regions in its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing and controls its speed to prevent overshoot.
Work on fuzzy systems is also being done in USA, Europe, China and India. NASA in USA has studied fuzzy control for automated space docking, as simulation showed that a fuzzy control system can greatly reduce fuel consumption. Firms such as Boeing, General Motors, Allen-Bradley, Chrysler, Eaton and Whirlpool have used fuzzy logic to improve on automotive transmission, energy efficient electric meters, low power refrigerators, etc.
Researchers are concentrating on many applications of fuzzy control systems, have developed fuzzy systems and have integrated fuzzy logic, neural networks and adaptive genetic software systems, with the ultimate goal of building self-learning fuzzy control systems.
This, in my opinion, is sufficient reason to induce you to start learning mathematics!
Geetha S. Rao,
Ex Professor, Ramanujan Institute for Advanced Study in Mathematics, University of Madras,
Chepauk, Chennai 600005.