Foundations of Mathematical Practice

The last (September 24, 2010) of several editions of a book prepared for use in a first-year undergraduate course called Fundamentals of Mathematics (MATH 111) at METU. Ten-point type, size A5 paper, 236 pages.

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The catalog description of MATH 111 at METU includes the following:

Content: Propositions. Connectives. Truth tables. Logical equivalencies. Logical implications. Methods of proofs of implications and equivalences. Quantifiers. The rules of inference for quantified propositions. Sets, subsets. Set operations. The laws of set theory. Cartesian product. Relations. Inverse relations. Composition of relations. Functions. Injective, surjective and bijective functions. Composition of functions. Equipollent sets. Countability of sets. More About Relations: Equivalence relations. Equivalence classes and Partitions. Quotient set. Order relations: Partial order, total order, well order. Mathematical Induction and Recursive definitions of functions.

Goals:

1. To enable the student to comprehend and construct mathematical arguments,
2. to develop the mathematical maturity of the student,
3. to provide basic definitions, facts and tools necessary for his further studies in mathematics.

The numbering of the Goals here is by me. I taught MATH 111 five times. In my view, Goal 1 and 2 should be goals of every course of mathematics. Moreover, these goals cannot be met by a course without real mathematical content. I attempted to give real mathematical content to the book above. This makes the book difficult—too difficult, perhaps, for a first-year course. And yet I think anybody who teaches a course like MATH 111 ought to know the contents of my book. Moreover, students should learn from teachers, and not only from books.

I think now that the parts of a course like MATH 111 that are needed in other courses ought simply to be taught in those other courses. The real mathematical content of MATH 111 is suitable for an upper-level course.