FOUNDATIONS OF MATHEMATICS (Part 1)

Mathematics includes many different subjects. So the term 'mathematics' is usually hard to define. But here is a definition that fits most of the mathematics taught in school or college:

MATHEMATICS is the study of quantities and relations through the use of numbers and symbols.

ARITHMETIC deals with quantities expressed by numbers.

ALGEBRA uses quantities and relations expressed by symbols.

GEOMETRY involves quantities associated with figures in space, such as length and area, and the relationships between figures in space.

TRIGONOMETRY is concerned with the measurement of angles and with the relationships of angles.

ANALYTIC GEOMETRY applies algebra to geometric studies.

CALCULUS works with pairs of associated quantities and the way one quantity changes in relation to the other.

SCIENCE depends on mathematics for exact descriptions and formulas of observations and experiments. In fact, there is no legitimate science without mathematics.

MATHEMATICS IS THE LANGUAGE OF SCIENCE

Without mathematics we would neither have modern technology nor modern medicine.

Mathematics, the language of science, is the language of dreamers who plan to achieve their dreams.

One can translate an idea into a set of drawings. To do this you can use symbols which are understood universally by people who turn dreams into realities.

Scientists of all languages exchange ideas by using the symbols of mathematics.

You need symbols not only to help you organize your own ideas, but also to explain your ideas to others.

FOUR FUNDAMENTAL CHARACTERISTICS OF SCIENCE

1.  Probably the first essential of a science is that it be objective and unbiased.

The scientist must get "the truth, the whole truth, and nothing but the truth," as the legal oath runs. When a scientist studies something, he or she must get correct facts about it, not sugared to suit someone's taste, and he or she must get all the relevant facts about it, not just a selection of the facts.

2.  A second characteristic of science is that it is interested in arriving at generalizations, usually about cause-and-effect relations.

Science is not interested in individual facts alone. The object of all science is the separation of what is common and general from what is accidental and different. For example, science is not interested in the fact that an apple falls to the ground when it is released from a height. Science is, however, interested in stating a relationship of attraction between the earth and other objects (including apples).

3.  A third characteristic of science is that it constantly develops better and better methods for securing the facts that are deemed necessary to arrive at the generalizations it seeks. 

4.  The fourth fundamental characteristic of a science is that science has achieved a coherent body of theoretical generalizations into which all new research is integrated. 

research is increasingly fruitful when it starts out with problems and hypotheses that are systematically interrelated and are based on previous research. 

NOTE:  THESE FOUR TRAITS ARE WHAT DISTINGUISHES SCIENCE FROM COMMON SENSE.

Everyone has common sense in a degree that depends on his or her native intelligence and the breadth of his or her experience. 

COMMON SENSE consists of generalizations drawn from personal experience, and is therefore partial and selective. 

But scientists also need specialized training and must acquire special habits of mind so that they will be objective, work toward generalizations not limited by personal experience, use methods and tools, and develop abstract theories. 

All these things can be learned just as any other habits and skills are learned. 

Scientists, like anyone else, are helped by common sense, since common sense usually helps them to learn the four types of habits and skills that scientists need, and it helps them to devise new tools for better observation. 

But scientists also have to be on their guard against common sense, since common sense is based on DIRECT experience whereas science must be based on CONTROLLED experience.
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RESOURCE: 'SOCIOLOGY: The Study of Human Relations' by Arnold M. Rose, UNIVERSITY OF MINNESOTA (1967)