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 irrelevant 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 students to think of a mathematical proof as some sort of meaningless ritual that they must learn to do simply because we require them to.  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 formalism 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 (Chapter 1).  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 (Chapter 2).

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 is like. So I have presented a set of axioms for the integers and proven a few elementary properties from them, so students can see the mental discipline required to set aside all our presumptions and work from the axioms alone.  But I think it is enough to have made that point.  So I go on to focus on the Well Ordering Principle as the property distinguishes the integers from other familiar number systems.

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.  But 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.  However I do think it is important that a text training students to develop and appreciate mathematical arguments not create the impression that careful proof is only important in elementary number theory or algebra.  They should see that it pervades all mathematics, analysis included.

Another goal is to train students to read more involved proofs such as they may encounter in advanced books 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 Chapter 5 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 anything they have encountered previously.  Chapter 6 gives a treatment of determinants.  The proofs of that chapter are mostly based on careful manipulations using the explicit formula for det(A), and provide an opportunity to help the students learn to scrutinize a detailed formula-based argument.  The final section develops a proof of the Cayley-Hamilton theorem.  This illustrates how a rigorous proof can emerge from a careful examination of a cute but questionable manipulation using the adjoint matrix.  Many people want their bridge course to involve ideas from linear algebra, and this chapter provides some of that.

I expect many of my colleagues to react with, "this is too hard for the typical student."  My philosophy is that the instructor has a rather different role to play than 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.  In this course especially, I view the instructor's role as helping students learn to read and work from a text written in a style typical of what they will encounter in their upper level courses.  So I have tried to write a book that is not a comfortable accommodation of where the students are when they start the course, but an example of the kind of exposition they will need to work from in the future, with their instructor serving as a coach for this their first encounter. 

Solutions

I have deliberately not made written solutions to the problems available in any form that can be easily posted on the web.  The availability of solutions online seems to be a temptation that few undergraduates can resist.  The result is to short-circuit the value of a text as a teaching tool.  I go over solutions on the board in class or in my office as needed, but I do not make solutions available in any electronic format.  If you are teaching from the book, I implore you to respect this restriction and not make solutions available in any electronic format.  If you are using the text for self-study I understand that prepared solutions would be valuable to you, but I will not provide them.  My advice is to seek out someone with experience in writing proofs and ask for their advice or help.

The Current Version

The current version is dated December 7, 2016.  It includes many typographical corrections and revisions suggested by students as well as those who have found the book online.  Thanks to all who have made suggestions.  If you find more I'll be grateful if you point them out to me.

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).  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.  Please note that under the no-derivatives restriction you may not distribute problem solutions in any electronic form.  For details of what the copyright allows, see the link in the copyright statement below.  -- M. Day