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A Logical Approach to Discrete Math |
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David Gries and Fred B. Schneider, have written a text
A Logical Approach to Discrete Math (Springer Verlag, 1993),which attempts to change how logic and discrete math is taught. Our thesis is that logic is the glue that binds together arguments in all domains. But this requires a logic that lends itself to formal application by people. For this purpose, we use a calculational logic, in which substitution of equals for equals rather than modus ponens is the main inference rule.
We discuss principles and heuristics developing proofs and work toward giving students a skill in formal manipulation. Thereafter, we use the logic in giving rigorous introductions to: set theory, mathematical induction, a theory of sequences, a theory of integers, functions and relations, combinatorics, solving recurrence relations, and modern algebra.
Our experience is that students are far more positive about notation, proof, and rigor with our treatment than they are after a conventional discrete math course. Many students say that they are now less apprehensive about mathematics and proof, and others say they are using their new skill in formal manipulation in other courses. Some teachers who have used our text (in primarily teaching institutions) say that the approach helps the weaker students more than the stronger ones. These experiences give us hope that adoption of our approach can lead over the years to a radical change in the field's attitude toward proof and mathematics and its ability to deal with formality.