fall 2007 course description
the mathematical perspective
homework guidelines
paper and talk guidelines
summary of proof techniques
additional resources
truth table forms
This course may be known at other schools as Introduction to Abstract Mathematics, Transition to Higher Mathematics, Bridge to Advanced Mathematics, etc. The objective of the course is to give students a chance to become more comfortable with the major elements of theoretical mathematics before they are asked to create proofs in content courses such as abstract algebra, real analysis, and topology. At Stetson, this course is a prerequisite for all upper level math courses.
The necessary topics are
Modern mathematics is based on the axiomatic system. Each area of mathematics begins with its own axioms (accepted, unproved statements) and primitive (accepted, undefined) terms. From these are built up theorems (proved statements) and defined terms. This organizational perspective dates back over two millenia to the Classical Greeks. But it has been used continuously only since the end of the 19th century, when logic and the foundations of mathematics were intensively studied. It is possible to learn and work within a field of mathematics without ever grasping this structure; that is the state of a student's knowledge throughout elementary and high school, and that was the state of the world for several thousand years of civilization. But to understand mathematics well, a wider view is now needed. As currently conceived and practiced, Mathematics is a tree whose foliage consists of various fields of study connected by branches of logical dependencies and history.
A student's first acquaintance with logical structure is usually in Euclidean geometry, where the axioms are made explicit, and the student may be asked to write proofs. It is rarely mentioned at this time that arithmetic and algebra, fields possibly more familiar to students, have their own axioms and theorems. The axioms of arithmetic are Peano's axioms, and those of algebra are the field properties (really theorems) of the real numbers. Living and working in a world of logical connections (as distinct from the world of formulas) requires a leap of the intellect.
I try to combine lecture, discussion, and student presentation of exercises and proofs. In some classes I have been able to use the Moore Method part of the time. After logic and set theory, we study the natural numbers beginning with Peano's axioms. I aim for a complete construction of the real numbers (via Dedekind cuts), although we rarely complete the whole trip. The idea is to give students a new look at mathematics, confidence in their ability to express themselves, and a good start on learning to construct their own proofs. As mentioned, this is a challenging task, which is why I chose content (arithmetic) that is familiar. Stetson math majors, on graduation, say this is one of the most valuable (albeit frustrating!) math courses they took.
The text I use is my own, Essentials of Mathematics, ©2003 MAA. In addition to the course material, it contains an introduction to mathematical culture. The sections on the number systems are based on a course I took at Eckerd College from George Lofquist, originally derived from Edmund Landau's Foundations of Analysis (Chelsea).
The two-valued system of logic used by mathematicians and taught to beginning math majors is no simple thing. When Russell and Whitehead published their Principia, the fields of logic and set theory were in upheaval from contradictions recently found. To give a hint of the problems, statements that followed the rules of grammar were found to be nonsensical: "This statement is false." And sets constructed by common-sense means were found to contain unacceptable contradictions: "The set of all sets." To get around the difficulties, many mathematicians worked for years to develop the currently accepted Zermelo-Frankel axioms of set theory, of which there are about nine (depending on author), not including the Axiom of Choice. The statements of these axioms are highly technical and largely ignored (though not violated) by mathematicians working in fields other than foundations.
Furthermore, there are other systems of logic: three-valued systems, many-valued systems, fuzzy logic, to name a few. Some of these have applications, and others seem for now to be just curiosities.
Lastly we mention Gödel's famous theorem. First, two definitions. A mathematical system is called consistent if its axioms lead to no contradictions. A mathematical system is called complete if every well-formed statement within the system can be proved either true or false. Gödel's theorem states that no mathematical system can be simultaneously consistent and complete. So mathematicians must choose between these two desirable characteristics.
fall 2007 course description
the mathematical perspective
homework guidelines
paper and talk guidelines
summary of proof techniques
additional resources
truth table forms
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