One day, I was reading a proof that I just couldn’t seem to understand. I understood the individual steps, but could not follow the overall logic of the proof. At this point, I realized that I had to resort to breaking down the proof into mathematical logic. When I reach this point, I go to my notes that I made while reading “How to Prove It” by Daniel Velleman. After writing down the logical structure of the proof and using various laws of logic I was able to see the reasoning behind the proof.
I raise this issue due to what appears to me to be a puzzling aspect of university mathematics education. Although many schools now offer a “foundations” course for new math majors in which they are taught mathematical logic, set theory, and proof structure and strategy, other schools do not. For those schools that do not, the assumption appears to be that students should figure out how to understand and write proofs by osmosis. Meaning that they will simply pick it up during an introductory analysis or linear algebra class. Clearly there are some students who were born to be pure mathematicians who will be able to accomplish this feat. However, I can’t imagine that such students are more than a very small percentage of math majors. Such a sink-or-swim attitude is difficult to understand and probably prevents many talented students from sticking with a math program.
If students do not have the tools to break down proofs into their basic logical structure, then what tools do they have to fall back on when they are having trouble understanding or constructing a proof? Thinking about my own experience that I related above, my knowledge of mathematical logic was indispensable for being able to work through my difficulties in understanding a proof.
Permit me to make plug for Velleman’s “How to Prove It”. This book starts with mathematical logic and then uses set theory to develop techniques for analyzing and writing proofs. What I like about this approach is that the actual mathematics is simple enough for a high school student to understand, so that it does not interfere with concentrating on the logical structure of proofs. This is a book that every prospective math major should read and master the summer before starting college.