What is Mathematics About?

(a) Introduction: Nature, Meaning and Definition of mathematics

Mathematics reveals hidden pattern that helps us to understand the world around us. Now much more than arithmetic, algebra and geometry mathematics today is a diverse discipline that deals with data, measurement and observation from science with inference, deduction and proof. Mathematics is an applied science. Many mathematicians focus their attention on solving problem that originates in the world of experience. Mathematics by nature is both pure, theoretical adventures of mind and a practically applied science. This dichotomy allows the theoretical mathematics to ''Do mathematics for mathematical sake'' and the applied mathematics to use mathematics as a tool to solve real problem'' Mathematics finds useful application in business, industry, music, politics, sport, medicine, agriculture, engineering and social and natural science. The result of mathematical theory and theorem are both significant and useful. Through its theorem mathematics offers science both a foundation of truth and a standard of certainty. In addition the theorem and theories offers distinctive models of thought which are versatile and powerful including modeling, abstraction, optimization, logical analysis, interference from data and use of symbols. Experience with mathematical models of thought builds mathematical power, a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to asses risk and to suggest alternatives. Mathematics empowers us to understand the world better. Many mathematicians have given their contribution in coming modern phase of mathematics. For example Euclid studied about geometry. Newton, Leibniz studied about calculus. Gauss, Joseph Fourier, Simeon Poisson, Augustine Louis Cauchy etc gave their contribution in Algebra and geometry. Some definitions given by the mathematics are presented below.

'' Mathematics is a way to settle in the mind a habit of reasoning." –Lock David Hilbert said, "Mathematics is nothing more than a game played according to certain simple rule with meaningless mark on paper." According as Russell, ''Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true." Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences." Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Wikipedia writes Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Although incorrectly considered part of mathematics by many, calculations and measurement are features of accountancy and arithmetic. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Being an open intellectual system, mathematics continued to develop, in fitful bursts, until the Renaissance, when mathematical innovations interacted with new scientific discoveries, leading to acceleration in research that continues to the present day. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Numerology is considered an application of mathematics by many but differs from mathematics in that it holds a mystical view of numbers. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.

There is strong relationship between science and mathematics. Science provides mathematics with interesting problem to investigate and mathematics provides science with powerful tools to use in analyzing the data. Also the mathematics is chief language of science. The symbolic language of mathematics has turned out to be externally valuable for expressing scientific idea unambiguously. We can list the nature of mathematics as follows:

Ø Mathematics is an inductive science

Ø Mathematics is a way of thinking

Ø Mathematics is an organized structure of knowledge

Ø Mathematics is an science and art both

Ø Mathematics is a language

Ø Mathematics is a study of patters.

As stated above mathematics can be categorized in to two parts.

(i) Pure Mathematics: 'Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigor, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on'-Wikipedia

(ii) Applied mathematics: 'Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modeling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data' -Wikipedia

(b) Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in our lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. There are mainly three types of philosophy of mathematics referred as logistic school of thought, intuitionist school of thought and formalist school of thought whose brief description are presented below:

(i) Logicism: Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. All mathematical concepts are to be formulated in term of logical concept. All the theorem of mathematics is to be developed as theorems of logic. Gottlob Frege was the founder of logicism however Dedikind, Bertrand Russell, Whitehead have also given their contribution on school of logic. Every theorem is of the form logic. So logic is the fundamental bases of mathematics. Without logic mathematics is no more remains mathematics. Logicism holds that logic is the proper foundation of mathematics, and that all mathematical statements are necessary logical truths. For instance, the statement "If Socrates is a human, and every human is mortal, then Socrates is mortal" is a necessary logical truth. To the logicist, all mathematical statements are precisely of the same type; they are analytic truths, or tautologies. Logicism is just the claim that the theorems of mathematics are logically necessary or analytic. Logicism does not belive on formality, and mathematical discovery. If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.

(ii) Intuitionism: Intuitionism is the immediate apprehension with out intervention of any logical process of knowledge of mental perception. Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived through mental construction that proves it to be true and the communication between mathematicians only serves as a means to create the same mental process in different minds. According as intuitionist philosopher mathematics is the production of human mind. From this school of thought we can say that mathematic is to be built solely by finite constrictive method on the intuitively given sequence of natural number. Thus Brouwer's intuitionism stands apart from other philosophies of mathematics; it is based on the awareness of time and the conviction that mathematics is a creation of the free mind, and it therefore is neither Platonism nor formalism. It is a form of constructivism, but only so in the wider sense, since many constructivists do not accept all the principles that Brouwer believed to be true. So intuitionism is no more like other philosophy of mathematics. It believes on the inner capacity of learner. How the individual perceive the knowledge is important than what they learn. Learning is the product of individuals mind.

(iii) Formalism: The formalist thesis is that mathematics is concerned with formal symbolic systems. If fact, mathematics is regarded as a collection of such abstract development, in which the terms are mere symbols and the statement are formulas involving these symbols are the ultimate base of mathematics. Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. Recently, some formalist mathematicians have proposed that all of our formal mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). It is usually said that formalist philosophy is the realist philosophy in present context. The intuitionism and logicism may not be sufficient of the learning of mathematics. To study about the geometry in class 10 the individual must study some basic knowledge about geometry in previous class. Every mathematical concept is in sequential form.