Mathematics

Mathematics is one of the best creations of human mind concerned chiefly with ideas, processes and reasoning. Mathematics is an integral part of all other forms of science such as Physics, Chemistry, Biology, Medicine and Engineering. All these Sciences progress only with the aid of Mathematics. Mathematics is necessary tool being employed by all other forms of Science. It would be wrong to consider Mathematics as a sum total of Trigonometry, Algebra, Statistics, Geometry, Arithmetic and Calculus. Mathematics is a way of thinking. It is a way of organizing a logical proof. Mathematics is not about answers, it's about processes. For more than two thousand years, mathematics has been a part of the human search for understanding. Mathematics discoveries are the result of the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning. It is said that Mathematics is the gate and key of the Science. According to the famous Philosopher Kant, "A Science is exact only in so far as it employs Mathematics". Since the last century, application of mathematics has been spreader over many areas like voting trends in politics, the dating of ancient artifacts, the analysis of automobile traffic patterns, and long-term strategies for the sustainable harvest of deciduous forests. Today mathematics is considered as a condition to call someone as a liberally educated person. It is said that all scientific education which does not commence with Mathematics is said to be defective at its foundation. Neglect of mathematics works injury to all knowledge. 

Origin/History

Steven Schwartzman in his book The Words of Mathematics wrote that Latin mathematica was a plural noun, which is why mathematics has an -s at the end even though we use it as a singular noun. Latin had taken the word from Greek mathematikos, which in turn was based on mathesis. That word, which was also borrowed into English but is now archaic, meant "mental discipline" or "learning," especially mathematical learning. The Indo-European root is mendh- "to learn." In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The roots of elemetary mathamatics is found in most of the Ancient civilizations. Greece, the Roman empire, Vedic society and ancient Egypt are some of them. However, The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1540. University of Aberdeen created the first chair of mathematics in 1613 followed by the Chair in Geometry being set up in University of Oxford in 1619. University of Cambridge in 1662 established the Lucasian Chair of Mathematics  . 

Trading, land measurement, painting and weaving patterns and the recording of time are some of the areas for which mathematics was used. The Babylonians and Egyptians began using complex mathematics like arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy around 3000 BC. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.

The earliest records of counting came from physical evidence. The oldest mathematical artifact was discovered in the mountains between South Africa and Swaziland. It is a piece of baboon fibula with 29 notches, dated 35,000 CB. Old Stone Age people had devised a system of tallying by groups as early as 30,000 BCE. The oldest record of primes is possibly the Ishango bone. Currently at the Musee d.Histoire Naturelle in Brussels has been dated about 6500 CB. Having three rows of notches, and one of the columns has 11, 13, 17, and 19 notches; It is believed that these notches formed a primitive calendar system.

Definitions

Mathematics has been defined in various ways since the primitive era. Some of the well-known definitions of Mathematics is mentioned below.

  “The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra”. Oxford English Dictionary

  “Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence.”- James Joseph Sylvester

  “Mathematics is the science that draws necessary conclusions.” - Benjamin Peirce

  “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas" -. G. H. Hardy

  “All Mathematics is Symbolic Logic.” -Bertrand Russell

Fields /Areas

The subject matter of mathematics is wide and it can broadly be divided in to the study of Quantity, Structure, Space and Change. Mathematics is also classified as in to Pure Mathematics and Applied Mathematics

What is Pure Mathematics?

Pure Mathematics is the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness. Following are the major fields of mathematics.  Pure mathematics explores the boundary of mathematics and pure reason. It has been described as "that part of mathematical activity that is done without explicit or immediate consideration of direct application," although what is "pure" in one era often becomes applied later. 

What is Applied Mathematics?

Applied mathematics is more practical in its applicability. It is closely associated with with mathematical methods that are typically used in science, engineering, business, and industry. Applied mathematics is often called as knowledge specific mathematics. As a profession applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Quantity: Numbers are the starting point to study quantity. Both natural numbers and integers with arithmetical interpretation is the core area of mathematical study of quantity. Theories like Fermat's Last Theorem, The twin prime conjecture Goldbach's conjecture are the outcome of deeper study of quantity. It is worth mentioning here that the study of size is closely associated with mathematical study of quantity.

Structure: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance. Mathematics studies properties of those studies properties of those sets that can be expressed in terms of that structure.(wiki)

Space Euclidean geometry in particular and geometry is general is the key focus area of study of space in mathematics. Quantity and space both are crucial in analytic geometry, differential geometry, and algebraic geometry. The study of space today generalizes the ideas related to space and numbers in Trigonometry that encompasses the well-known Pythagorean theorem.

Change Calculus was primary developed as a powerful tool to investigate understands and describe changes in the natural sciences. Functions play a vital role here as it is a primary concept here to describe a changing quantity. Functional analysis pays close attention on spaces of functions. Quantum mechanics is one of many applications of functional analysis.

Importance of Mathematics

Galileo: "The great book of nature can be read only by those who know the language in which it was written. And that language is mathematics."

Mathematics is the way to understand all sorts of things in the world around us. Mathematics is considered as the gate and key of the Science. As noted by Kant, "A natural Science is a Science in so far as it is mathematical".  Mathematics is interesting and challenging. It is always exciting to find a solution for a problem. Mathematics brings this excitement and sharpness in thought process making a useful real life experience. And Mathematics has played a very important role in building up modern Civilization by perfecting all Science. The utility of modern day mathematics can be understood with the help of following points:

• The mathematics of error-correcting codes is applied to CD players and to computers.

• Whenever it is said that advances are made with supercomputers, there has to be a mathematical theory which instructs the computer what is to be done, so allowing it to apply its capacity for speed and accuracy.

• The development of computers was initiated in this country by mathematicians and logicians, who continue to make important contributions to the theory of computer science.

• The next generation of software requires the latest methods from what is called category theory, a theory of mathematical structures which has given new perspectives on the foundations of mathematics and on logic.

• The physical sciences (chemistry, physics, oceanography, astronomy) require mathematics for the development of their theories.

• In ecology, mathematics is used when studying the laws of population change. 

• Statistics provides the theory and methodology for the analysis of wide varieties of data.  Statistics is also essential in medicine, for analyzing data on the causes of illness and on the utility of new drugs. 

• Travel by aero plane would not be possible without the mathematics of airflow and of control systems. 

• Body scanners are the expression of subtle mathematics, discovered in the 19th century, which makes it possible to construct an image of the inside of an object from information on a number of single X-ray views of it. Thus mathematics is often involved in matters of life and death.

• Voyager's journey to the planets could not have been calculated without the mathematics of differential equations.

Mathematics as a Career

Mathematics prepares you for careers in many fields. Every successful student in mathematics is fortunate enough to have a wide range of career choices. Mathematics develops multiple skills in a person. To name them:

a) Ability to think logically

b) Problem solving skills

c) Ability assume practically about something

d) Promptness etc.

Career opportunities are available for a mathematics graduate or a post graduate across all the employment sectors.  As per a survey in 2009 it is revealed that the top three best jobs in terms of income and other factors were careers suited for math majors. Another recent survey shows that the top 15 highest-earning college degrees have a common element: mathematics.  Following is a list of jobs associated with mathematics with a brief description of the job role:

• Actuary-- assemble and analyze statistics to calculate probabilities of death, sickness, injury, disability, unemployment, retirement, and property loss; design insurance and pension plans and ensure that they are maintained on a sound financial basis.

• Mathematics teacher-- introduce students to the power and beauty of mathematics in elementary, junior high, or high school mathematics courses.

• Operations Research Analyst-- assist organizations (manufacturers, airlines, military) in developing the most efficient, cost-effective solutions to organizational operations and problems; this includes strategy, forecasting, resource allocation, facilities layout, inventory control, personnel schedules, and distribution systems.

• Statistician-- collect, analyze, and present numerical data resulting from surveys and experiments.

• Physician-- diagnose patient illnesses, prescribe medication, teach classes, mentor interns, and do clinical research; students with a good mathematics background will find themselves being admitted to the best medical schools and discover that mathematics has prepared them well for the discipline, analysis, and problem- solving required in the field of medicine.

• Research scientist-- model atmospheric conditions to gain insight into the effect of changing emissions from cars, trucks, power plants, and factories; apply these models in the development of alternative fuels.

• Computer scientist-- interface the technology of computers with the underlying mathematical principles of such diverse applications as medical diagnoses, graphics animation, interior design, cryptogrraphy, and parallel computers.

• Inventory Strategist-- analyze historical sales data, model forecast uncertainty to design contingency plans, and analyze catalog displays to make them more successful; analyze consumer responses.

• Staff systems air traffic control analyst-- apply probability, statistics, and logistsics to air traffic control operations; use simulated aircraft flight to monitor air traffic control computer systems.

• Cryptologist-- design and analyze schemes used to transmit secret information.

• Attorney-- research, comprehend, and apply local, state, and federal laws; a good background in mathematics will help a student get admitted to law school and assist in the understanding of complicated theoretical legal concepts.

• Economist-- interpret and analyze the interrelationships among factors which drive the economics of a particular organization, industry, or country.

• Mathematics professor-- teach mathematics classes, do theoretical research, and advise undergraduate and graduate students at colleges and universities.

• Environmental Mathematician-- work as member of interdisciplinary team of scientists and professionals studying problems at specific Superfund sites; communicate effectively across many academic discilplines and be able to summarize work in writing.

• Robotics Engineer-- combine mathematics, engineering, and computer science in the study and design of robots.

• Geophysical Mathematician -- develop the mathematical basis for seismic imaging tools used in the exploration and production of oil and gas reservoirs.

• Design -- use computer graphics and mathematical modeling in the design and construction of physical prototypes; integrate geometric design with cost-effective manufacturing of resulting products.

• Ecologist -- study the interrelationships of organisms and their environments and the underlying mathematical dynamics.

• Geodesist -- study applied science involving the precise measurement of the size and shape of the earth and its gravity field (courtesy of Bruce Hedquist).

• Photogrammetrist -- study the applied science of multi-spectral image acquisition from terrestrial, aerial and satellite camera platforms, followed up by the image processing, analysis, storage, display, and distribution in various hard-copy and digital format (courtesy of Bruce Hedquist).

• Civil Engineer -- plan, design, and manage the construction of land vehicle, aircraft, water, and energy transport systems; analyze and control systems for land vehicular traffic; analyze and control environmental systems for sewage and water treatment; develop sites for industrial, commercial and residential home use; analyze and control systems for storm water drainage and storage; manage construction of foundations, structures and buildings; analyze construction materials ; and surface soils and subterranean material analysis (courtesy of Bruce Hedquist).

• Geomatics Engineer -- once known as "surveying engineer", includes geodetic surveying : takes into account the size and shape of the earth, in order to determine the precise horizontal and vertical positions of geodetic reference monuments; cadastral surveying.