This paper answers four questions about the nature of mathematics.
1. What is needed first is a definition of mathematics that is broad enough to cover all mathematics taught up through high school, but specific enough to exclude other fields, such as biology, music, and social studies -- although the latter can have mathematical aspects. The fields of mathematics to be covered include arithmetic, algebra, and geometry, and, in the best high schools, trigonometry, analytic geometry, and calculus.
All these fields except for plane geometry deal primarily with numbers, and even geometry is brought within the numerical domain by means of analytic geometry. However, there are some types of mathematics that deal not with numbers, but with logical relationships, as in symbolic logic and the use of Venn diagrams.
One can look at the relationship among the fields of mathematics the other way round as well. The concept of a logical proof that one learns in geometry applies to all other mathematics as well. It can be proved that the correct solution to an arithmetic or algebra problem is the only logical solution.
In other words, when one is dealing with some type of mathematics, the correct answer to a question is not a matter of opinion. If everyone is agreed on the meaning of the terms in the question and on the correctness of the calculations or reasoning, then everyone has to agree on the correctness of the answer. In order for all to agree on all this, the meanings of the terms must be completely unambiguous; hence, words, which are inherently ambiguous, cannot be adequate for this purpose. Instead, mathematics deals with numbers or with other concepts that can be defined precisely.
In this sense, one could offer the following as a definition of mathematics: whenever one is dealing with a question to which the answer is a matter of logic, not opinion, then one is dealing with mathematics. Even within fields that a...