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Andrew Wohlgemuth

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Abstract/ Summary

Mathematics has two fundamental aspects: (1) discovery/logical deduction and (2) description/ computation. Discovery/deductive mathematics asks the questions:

1. What is true about this thing being studied?

2. How do we know it is true?

On the other hand, descriptive/computational mathematics asks questions of the type:

3. What is the particular number, function, and so on, that satisfies ... ?

4. How can we find the number, function, and so on?

In descriptive/computational mathematics, some pictorial, physical, or business situation is described mathematically, and then computational techniques are applied to the mathematical description, in order to find values of interest. The foregoing is frequently called “problem solving”. Examples of the third question such as “How many feet of fence will be needed by a farmer to enclose ...” are familiar. The fourth type of question is answered by techniques such as solving equations, multiplying whole numbers, finding antiderivatives, substituting in formulas, and so on. The first two questions, however, are unfamiliar to most. The teaching of computational techniques continues to be the overwhelming focus of mathematics education. For most people, the techniques, and their application to real world or business problems, are mathematics. Mathematics is understood only in its descriptive role in providing a language for scientific, technical, and business areas.

Mathematics, however, is really a deductive science. Mathematical knowledge comes from people looking at examples, and getting an idea of what may be true in general. Their idea is put down formally as a statement—a conjecture. The statement is then shown to be a logical consequence of what we already know. The way this is done is by logical deduction. The mathematician Jean Dieudonne has called logical deduction “the one and only true powerhouse of mathematical thinking”. Finding proofs for conjectures is also called “problem solving”. The “Problems” sections of several mathematics journals for students and teachers involve primarily problems of this type.

The deductive and descriptive aspects of mathematics are complementary—not antagonistic—they motivate and enrich each other. The relation between the two aspects has been a source of wonder to thoughtful people.


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