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

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ughts be said in any realistic way to be doing mathematics? Perhaps this is parallel to the question about whether a tree falling in the forest makes any sound if no one is around to hear it. Further, if the emphasis in this definition is on thinking, then the issue is raised of whether mathematicians think quantitatively, in contrast to everyone else, who would thus be thinking, say, qualitatively. But here one has wandered into the realm of psychology, in which the ôrightö answers are largely a matter of opinion. It would seem illogical to have a definition of mathematics whose correctness is merely a matter of opinion, rather than one that is logically self-evident. 4. Bullock (1994) argues that mathematics is essentially a language in its own right, one whose peculiarities result from the fact that its objects of discourse are invented, not discovered. That is, there is no ambiguity in the mathematical description of a sphere, because the properties of a sphere are all defined; it is an imaginary object, not a physical one. He then makes the point that, when applied to real objects, mathematics is metaphorical, saying, ôIt is just as metaphorical to call the world a sphere as it is to call it a stageö (583). Bullock say
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