BRANCH OF MATHEMATICS
Applied Mathematics
In a classroom much of the mathematics we teach is applied mathematics in the sense that it relates directly to life’s activities connected with buying, selling, trade, business, consumer applications,weighing, measuring etc. These applications of mathematics to the world around us can be extended to more technical ones.Mathematics has helped in analysing motion and in doing so, Newton created the calculus which became known as applied mathematics.
More recently mathematical growth has been in areas such as operational research, linear programming, system analysis, statistics, all involving processes to handle numerical information in an increasingly technologically advanced world. The mathematical ideas we teach in schools develop over many years of study and become associated in our minds with all the applications and illustrations presented to explain them. It is always easier to explain what we can do with a concept in mathematics than to say what it is. A teacher has to answer questions such as “what is the use of this to us this?’, or “why do we have to learn this?’If he/she fails to do so there will be many children who will not be able to see the point.
Pure Mathematics
In pure mathematics we start from certain rules of inference, by which we can infer that if one proposition is true, then some other proposition is also true. These rules oi inference constitute the major part of the principles of formal logic. For instance, we all know the axiom that in real numbers if a > b and b > c, then a > c. Thus, from given propositions we conclude that some other proposition is true. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some (one or more) particular person or thing, then our deductions constitute mathematics. Thus, mathematics may be defined as “the subject in which we never know what we are talking about, nor whether what we are saying is me” – Bertrand Russell
These ideas point out the abstract nature of mathematics. Mathematics deals with the application of arbitrary rules in an arbitrary situation which may or may not have significance in the world outside. It is a network of logical relationships. In school mathematics Euclidean geometry is essentially pure mathematics. A set of axioms and postulates are given and from them a body of definitions, theorems and propositions are derived. All pure mathematics is built up by combinations of primitive ideas of logic; its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference.
There is a very thin line dividing pure and applied concepts. On the one hand concepts of pure mathematics are formulated because of the need to apply them and on the other, every discovery or formulation has some application somewhere.
MEANING AND NATURE OF MATHEMATICS
Meaning of Mathematics
The term Mathematics has been interpreted and explained in various ways. This is due to our empirical knowledge and various types of experiences. But ultimately all types of explanations conclusively end at some kinds of relationship with number and space. Thus, mathematics deals with quantitative facts and relationships as-well-as with problems involving space and form.
It enables the man to study various phenomena in space and establish different types of relationship between magnitudes of quantitative and quantitative facts. Therefore it may be concluded that mathematics is the enumerative and calculative part of human life and knowledge the person to given an exact interpretation.
According to New English Dictionary,“Mathematics, in a strict sense, is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations.”
According to Webster’s Dictionary, “Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations and abstractions and of space configuration and their structure, measurement, transformations and generalizations.”
In the words of Locke, “Mathematics is a way to settle in the mind a habit of reasoning.”
According to Roger Bacon, “Mathematics is the gate and key of the sciences…. Neglect of mathematics work injury to all knowledge, since who is ignorant of it cannot know the other sciences or the things of the world. And what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy.”
“On the basis of above definitions, we can say or conclude that ”
- Mathematics is the science of magnitude and numbers.
- Mathematics is the science of quantity and space.
- Mathematics is the important means of generalisation
- Mathematics is the an applied science for the expression of other sciences.
- Mathematics is the method of progress of various subjects.
- Mathematics as the means to draw conclusion and judgement.
- Mathematics is the perfection of generalisation.
- Mathematics is the science of logical reasoning.
Nature of Mathematics
What is Mathematics and how does it grow are the basic questions which all the students of Mathematics must understand. In school, those subjects which are included in the curriculum must have certain aims and objectives on the basis of which its nature is decided. Mathematics holds a strong and unbreakable position as compared to other school subjects.
With this reason, mathematics is more stable and important than other school subjects. The way in which the structure of a subject becomes weak, its truthfulness, reliability and prediction also decreases in the same manner. On the basis of this specific structure, the nature of each subject is determined and placed in the school curriculum.
It is not necessary that all subjects have same nature. Mathematics has its unique nature thus on the basis of which we can compare it with other subjects. The basis of comparison of two or more subject is their nature. We can understand the nature of Mathematics on the basis of following features
- Mathematics is a science of space, numbers, magnitude and measurement.
- Mathematics has its own language. Language consists of mathematical terms, mathematical concepts, formulae, theories, principles and signs, etc.
- It gives accurate and reliable knowledge.
- Mathematics knowledge is exact, systematic, logical, and clear so that once it is captured it can never be forgotten.
- Mathematical rules, laws and formulae, are universal and that can be verified at any place and time.
- It develops the ability of induction, deduction and generalisation.
- Mathematics helps in developing scientific attitude among children.
- The study of Mathematics gives the training of scientific method to the children.
- Mathematical knowledge is based on sense organs.
- Mathematics is a systematised, organized and exact branch of science.
- Mathematics involves conversation of abstract concepts into concrete form.
- Mathematics is the science of logical reasoning.
- Mathematics does not leave any doubt in the mind of learner about theories, principles concepts etc.
- Mathematics helps to develop that habit of self-confidence and self reliance in children.
- Mathematics helps in the developments of sense of appreciation among children.
- Mathematics language is well defined, useful and clear.
- It draws numberical inferences on the basis of given information and data.
- Mathematical knowledge is applied in the study of science and in this different branches; for example physics, chemistry, biology, and other sciences.
- It is not only useful for different branches of science but also helps in its progress and organisation.
Thus on the basis of above points we can understand the nature of mathematics and draw conclusion that the structure of mathematics is indeed the basis of its nature and is more strong as compared to other school subjects. That is why its study is essential in school education.
MATHEMATICS AS THE SCIENCE OF LOGICAL REASONING
Reasoning is based on previous established facts. To establish a new fact or truth one has to put it on test of reasoning. If the new fact coincides with the previously established facts, it is called logical or rational. Logical reasoning is beyond subjectiveness.
In the process of logical reasoning, we approach everything with a question mark in our mind. For each question we make a hypothesis and this hypothesis is tested empirically or theoretically with the help of previously proved or established truths or facts. In mathematical working we also move upwards by the process of reasoning.
From our observation of physical and social environment we form certain intuitive ideas or notions called postulates and axioms. These postulates and axioms are self-evident truths and need no further proof or explanation. Thus, postulates and axioms are assumed to be true without reasoning. But this does not mean that here we ignore the process of reasoning. Actually self-evident truths are beyond reasoning. That is why we can not assume any evidence to be true. Only those evidences can be assumed as true that could not be proved untrue or irrational by existing logical knowledge.
Thus, postulates and axioms are bases of mathematics as-well-as of our process of logical reasoning. In mathematics we make several propositions and while proving a proposition we base our arguments on previously proved proposition. Thus, each proposition is supported by another proposition that has already been proved or established. Consequently if we go back one-by-one, we reach to a propositions that is based on postulates and axioms. Thus, in mathematics we always use the process of logical reasoning. Therefore, mathematics may be called as the science of logical reasoning.
In mathematics two types of reasoning is used. These prominent types of reasoning are:
Inductive Reasoning
Deductive Reasoning
“Mathematics in the making is not a deductive science, it is an inductive, experimental science and guessing is the tool of mathematics. Mathematician like all other scientists, formulate their theories form bunches, analogies and simple examples. They are pretty confident that what they are trying to prove is correct, and in writing these, they use only the bulldozer of logical deduction”
Whitehead has also emphasised the importance of deductive reasoning in mathematics by saying, “Mathematics in its widest sense is the development of all types of deductive reasoning.”
D’ Alembert says, “Geometry is a practical logic, because in it, rules of reasoning are applied in the most simple and sensible manner.
Pascal says, “Logic has borrowed the rules of geometry. The method of avoiding error is sought by everyone. The logicians profess to lead the way, the geometers alone reach it, and aside from their science there is no true demonstration.”
Geometry is a true demonstration of logic Mathematics is the only branch of knowledge, in which logical reasoning or logical laws are applied and the results can be verified by the method of logical reasoning.
W.C.D. Whetham– “Mathematics is but the higher development of Symbolic Logic.”
C.J. Keyser- “Symbolic Logic is Mathematics; Mathematics is Symbolic Logic.”
“The symbols and methods used in the investigations of the foundation of mathematics can be transferred to the study of logic. They help in the development and formulation of logical laws. In mathematics the symbol has got a meaning, e.g., a < b means ‘a’ is less than ‘b’. In logic, the meaning of this symbol has been extended. Let ‘a’ denote the class denoted by the cows and ‘b’ stand for the class denoted by the animals then a < b is easily interpreted to mean “a is included in b”, that is, all cows are animals.
For another example, take the symbol ‘x’. Let A denote the class. “Teachers’ and B the class, ‘Ladies.’ AXB may be interpreted to mean the class of persons who are both Teachers and Ladies.
Thus the meanings of mathematical symbols have been extended to represent the relationship of propositions in logic. The aims of the mathematician and those of the logician are practically the same.
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