This page contains some material related to the course MA197x, Bridge to Higher Mathematics. This course is a significant revision of the old MA196x, Knowing with Certainty: Proofs in Contemporary Mathematics.
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Text: Write Your Own Proofs in Set Theory and Discrete Mathematics by Amy Babich and Laura Person.
This course is an introduction to mathematical thinking aimed not only at the beginning mathematics, statistics or actuarial major, but also at students seeking to further their mathematical interests and those simply curious about logic and reason.
The course has two aims. The students in the course will be expected to explain, justify, defend, disprove, conjecture and verify, both verbally and in writing, mathematical ideas. One expected by-product of this training is that the student will develop concrete proof-writing skills which will enhance his or her chances for success in more advanced mathematics courses.
The second goal, for which we will have less time, is to survey the role of proofs and related constructs in contemporary mathematics and other parts of science. We will discuss topics such as computer-generated proofs, logic programming, the role of proofs in physics, the Kepler Conjecture, and Fermat's Last Theorem. Regular classroom discussion will draw the connection between the concrete problem-solving theme of the assignments and the more philosophical context in which most projects will be carried out.
A more detailed rationale appears below.
Prerequisites: Some college-level mathematics
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Unfortunately, this leads too many good students to the conclusion that they "don't like proofs" or "can't do proofs".
Over the years, the Department of Mathematical Sciences has taken various steps to help solve this problem. We designated five 2000-level math courses as "transition courses". These courses --- with course numbers 2073, 2271, 2273, 2431 and 2631 --- have as their main goal the introduction of some interesting and useful mathematical subject, but also as a subtext, a goal to introduce the student to logical thinking, to concepts like the contrapositive, quantifiers, mathematical induction, counterexamples and so on. These courses are very popular, but some students take them too late, or don't get enough of the foundational material out of the subtext to prepare them to be successful in Advanced Calculus (MA3831, MA3832) or modern algebra (MA3823, MA3825). Another component to the department's approach to this issue was the creation of the "analysis sequence", MA1031 through MA1034. MA197x offers another point of entry. Students who love math will love doing math at the higher levels, if only they are properly prepared for the new ways in which mathematics is done.
The course should also be fun! It would be nice if the course could bring students to the cutting edge of research, but one must be realistic about how much one can accomplish with only minimal exposure to university-level mathematics. So the compromise is to include a sort of "current events" component where students will learn about some of the latest results in mathematics and still have manageable proof-oriented homework assignments.
Students will find Professor Martin a tough grader, who expects students to not only work out the answer to a problem but to explain it in a simple, clear and efficient way, using proper English sentences. All who enter the course should be forewarned that this is not a cake walk!