CTET 2015 Exam Notes : TEACHING OF MATHEMATICS
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:
“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.