Реферат: Mathematics as a science

Mathematics as a science, viewed as a whole, is a collection ofbranches.

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

Arithmetic, algebra, the study of functions, the calculus,differential equations, and various other subjects which follow the calculus inlogical order are all developments of the real number system.

This part of mathematics is termed the mathematics of number.

A second branch is geometry consisting of several geometries.

Mathematics contains many more divisions.

Each branch has the same logical structure: it begins with certainconcepts, such as the whole numbers or integers in the mathematics of number,and such as point, line and triangle in geometry.

These concepts must verify explicitly stated axioms.

Some of the axioms of the mathematics of number are theassociative, commutative, and distributive properties and the axioms aboutequalities.

Some of the axioms of geometry are that two points determine aline, all right angles are equal, etc.

From the concepts and axioms theorems are deduced.

Hence, from the standpoint of structure, the concepts, axioms andtheorems are the essential components of any compartment of mathematics.

We must break down mathematics into separately taught subjects,but this compartmentalization taken as a necessity, must be compensated for asmuch as possible.

Students must see the interrelationships of the various areas andthe importance of mathematics for other domains.

Knowledge is not additive but an organic whole and mathematics isan inseparable part of that whole.

The full significance of mathematics can be seen and taught onlyin terms of its intimate relationships to other fields of knowledge.

If mathematics is isolated from other provinces, it losesimportance.

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