An Introduction to Proofs and the Mathematical Vernacular

by Martin Day

The typical university calculus sequence, which serves majors in the physical sciences and engineering as well as mathematics, emphasizes calculational technique.  In upper level mathematics courses, however, students are expected to operate at a more conceptual level, in particular to produce "proofs" of mathematical statements.  To help students make the transition to more advanced mathematics courses many university mathematics programs include a "bridge course" .  Many texts have been written for such a course.  I have taught from a couple of them, and have looked at numerous others. These various texts represent different ideas for what a bridge course should emphasize.  Not having found a text that was a good fit with my own ideas, I decided to try to write one of my own.  I am making the book freely available; a link is provided at the bottom of this page.  But first I want to explain the ideas which I have tried to embody in the book.

My Philosophy for the Bridge Course.

The students taking this course have (I assume) completed a standard technical calculus sequence.  They will have seen some proofs, but may have dismissed them as not relevant to what they needed to know for homework or exams.  We now want them to start thinking in terms of properties of mathematical objects and logical deduction, and to get them used to writing in the customary language of mathematics.  I don't think we accomplish that with the how-to approach to writing proofs that some texts take.  That encourages them to think of a mathematical proof as some sort of meaningless ritual or dance that they must learn to do.  Rather we want them to begin to think like mathematicians, and to become conversant with the language of written mathematics.

One of my disappointments with existing textbooks is that they often begin with too much formalism about propositional logic.  My experience is that whatever students learn from that is left by the wayside as soon as they move into a mathematical context of any substance.  My premise is that one learns precise logical language in the context of a real mathematical discussion, not from a "content-free" formal summary of logical grammar.  So rather than starting with logical form in the absence of substance, I start with some substance.  Specifically the first chapter simply jumps in with some proofs.  Then, with those proofs as examples, we can discuss how they are structured logically and talk about the language with which they are written.

Another concern I have with some texts is their deconstructive approach.  Students are implicitly told to forget what they know, because we want to start from scratch with an axiomatic approach.  For instance if we develop the integers or real numbers from their axioms, we have to ask the students to suspend what they already know about these basic number systems so that we can develop them anew from the axioms. Instead of building our students' knowledge we seem to be dismantling it and sending them backwards to more primitive topics.  I want to downplay that and instead develop topics that are not so obvious to the students, so that when we prove something we are moving forward rather than backward.  I do think it is important for students to understand what a set of axioms is, and what an axiomatic development it like.  So I have presented a set of axioms for the integers and a few proofs based on them so they can see the mental discipline required to set aside all our presumptions and work from the axioms alone.  But having made that point, I bring that discussion to a close and expressly return to reliance on our innate knowledge of the integers and their properties.

The students in this course have finished two years of calculus and related material.  Many bridge course texts do not touch on that material at all.  It is my desire to incorporate at least some problems and examples that employ ideas and techniques from differential calculus, in addition to the usual topics such as the Euclidean algorithm and modular arithmetic.  Analysis is very rich in content, which makes for many opportunities for creativity in developing arguments.  Most students are not very adept at using the ideas of calculus yet, and probably will go on to an advanced calculus course after this one, so I keep use of analysis relatively simple.  But I do think it is important that a text training students to develop and appreciate mathematical arguments and connections not create the impression that careful proof is only important in elementary number theory or algebra.  They should see that it pervades all mathematics.

Another goal is to train students to read more involved proofs such as they may encounter in textbooks and journal articles.  This involves being able to fill in details that a proof leaves to the reader.  Even more important is being able to look past the details to see the fundamental idea behind a proof.  To this end the final chapter is built around some results about polynomials (Descartes' Rule of Signs and the Fundamental Theorem of Algebra) whose proofs are accessible to students at this level, but are more substantial than what they have encountered previously.

I expect many of my colleagues to react with, "this is too hard for the typical student."  My philosophy is that the instructor in a course has a role to play in addition to the written text.  He/she does not merely recite the material and grade papers, but serves as a sort of intellectual trainer, prodding students toward more sophisticated points of view and encouraging them in the face of new challenges.  The book should not be a comfortable accommodation of where they are when they start the course, but should exhibit the level of sophistication to which we (as instructors) hope they will rise with our coaching.  It should be a goal to strive for, not a merely a platform to start from.

The Current Version

The first version, dated January 6, 2008, was taught through in spring 2008.  As expected it was very rough, with many typos and errors.  The current version, dated July 22, 2008, has corrected those, but also includes a number of revisions suggested by the students and my own judgment of what was and was not effective.  Additional revisions are likely in the future.

Downloading the Book

You are welcome to download the current version of the book (pdf file), use it,  and redistribute it for noncommercial purposes (such as provide it to students, either electronically or by having your local copy shop print it up for them).  For details of what the copyright allows, see the link in the copyright statement below.  If you do use the book to teach a course, I would enjoy to hear from you about how it worked out and any comments or suggestions you have.  -- M. Day