Deduction and Induction

The scientists have proved a chain of theorems and have come to recognize the entire structure of undefined terms, definitions, assumptions, and theorems as constituting an ab­stract logical system. In such a system we say that each proposition is derived from its predecessor by the process of logical deduction. This process of logical deduction is scientific reasoning.

This scientific reasoning must not be confused with the mode of thinking employed by the scientist when he is feeling his way toward a new discovery. At such times the scientist, curious about the sum of the angles of a triangle, proceeds to measure the angles of a great many triangles very carefully. In every instance he notices that the sum of the three angles is very close to 180°; so he puts forward a guess that this will be true of every triangle he might draw. This method of deriving a general principle from a limited number of special instances is called induction.

The method of induction always leaves the possibility that further measurement and experimentation may necessitate some modification of the general principle. The method of deduction is not subject to upsets of this sort.

When the mathematician is groping for (ищет) new mathematical ideas, he uses induction. On the other hand, when he wishes to link his ideas together into a logical system, he uses deduction. The laboratory scientist also uses deduction when he wishes to order and classify the results of his observations and his inspired guesses and to arrange them all in a logical system. While building this logical system he must have a pattern (модель) to guide him, an ideal of what a logical system ought to be. The simplest exposition (изложение) of this ideal is to be found in the abstract logical system of demonstrative geometry.

It is clear that both deductive and inductive thinking are very useful to the scientist.

Ex. 18. Writing. Put the sentences into the right order to make a complete paragraph. The first sentence is given to you.

WHAT IS MATHEMATICS?

1. Maths, as science, viewed as a whole, is a collection of branches.

A. The largest branch is that which builds on ordinary whole numbers, fractions, and irrational numbers, or what is called collectively the real number system.

B. Hence, from the standpoint of structure, the concepts, axioms and theorems are the essential components of any compartment of maths.

C. These concepts must verify explicitly stated axioms. Some of the axioms of the maths of numbers are the associative, commutative, and distributive properties and the axioms about equalities.

D. Arithmetic, algebra, the study of functions, the calculus differential equations and other various subjects which follow the calculus, in logical order are all developments of the real number system. This part of maths is termed the maths of numbers.

E. Some of the axioms of geometry are that two points determine a line, all right angles are equal, etc. From these concepts and axioms, theorems are deduced.

F. A second branch is geometry consisting of several geometries. Maths contains many more divisions. Each branch has the same logical structure: it begins with certain concepts, such as the whole numbers or integers in the maths of numbers or such as points, lines, triangles in geometry.

UNIT 2

“Mathematics is the gate

and key to science”

Roger Bacon