Foundations of Mathematics Text
My Foundations of Mathematics text was published by Wiley in March 2008. Answers to all the problems will be available to instructors adopting the text through the publisher. Here is a link to the publisher's page: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470085010.html.
Foundations of Mathematics
By Thomas Q. Sibley
This text will provide sophomore level mathematics majors the understanding and skills necessary to read and think about mathematics and write proofs. But it seeks to do more than simply compete with the many current texts. I intend to apply the most successful lessons from the “calculus reform” movement to this proof oriented course.
The graphical, symbolic and numerical reinforcement of concepts in reform calculus texts have helped calculus students’ intuition and problem solving ability. These texts’ conceptual questions push students beyond computing and symbol manipulation without jumping to general abstract questions or highly symbolic questions. Sophomore mathematics majors emerging from problem solving courses would correspondingly benefit from a deeper intuition about definitions and abstract concepts. They will then be better able to read mathematical text, formulate proofs and answer theory based questions. My text will facilitate this transition through the ordering of topics and through the explanations, examples and problems. While I will build intuition, I will not lower the ultimate expectation that students understand mathematics and write proofs at the level of successful mathematics majors.
An increasing number of mathematics departments have responded to the difficulties typical mathematics majors find in upper division courses by requiring a course introducing proofs and basic mathematical structures. While many texts already address this need, they suffer from various deficiencies. Almost universally, these texts assume that students can absorb new concepts and definitions more quickly than, in my experience, average majors can. For example, these books expect students to use a concept in proofs in the section introducing the concept.
These texts have various other shortcomings as well. Most neglect to explain how to understand mathematical definitions and how to read mathematical texts. They too often use colloquial wording for definitions, theorems and problems that weaker students find misleading. Finally some texts do not explain sufficiently clearly how to generate and write proofs.
I am confident I can write a superior text, one that helps students understand mathematical language, develop their intuition and write proofs. I have learned a great deal about writing texts from writing first an unpublished liberal arts mathematics text and subsequently The Geometric Viewpoint: A Survey of Geometries, published by Addison, Wesley and Longman (1998). In addition, my graduate training in mathematical logic and long years of teaching mathematics have honed my ability to explain logical language and proof formats. I have learned how to build students’ intuition, enabling me to introduce concepts in an understandable way. Reviewers of my geometry text noted how well its innovative problems built students’ intuition.
Students often need help making the transition from problem based courses into proof based courses, especially abstract algebra and analysis. A foundations or transition course is the most common response to this need. While the better mathematics majors readily make this transition on their own, many mathematics departments count on foundations courses to help more students succeed.
The students taking a foundations course have generally had some problem based college courses—two or three semesters of calculus and perhaps linear algebra or differential equations—usually with little or no emphasis on proof. Since many high schools have minimized the amount of proof taught in their geometry classes, a foundations course may well be a student’s first encounter with mathematical proof. Further, in problem solving courses, students often do not read the text, let alone understand or use formal mathematical definitions. Indeed, many calculus texts muddy the waters by labeling intuitions—e.g. the derivative is the slope of the tangent—as definitions. Thus students often experience mathematical definitions as oddly worded notation, rather than exceptionally clear and concise keys to proving results. Since they need to learn proof formats and devise proof strategies at the same time as they wrestle with new concepts, they encounter significant frustration. In fact, I think it speaks well for the skill and dedication of mathematics faculty that so many students succeed in foundations courses.
The faculty teaching foundations courses are generally Ph.D. mathematicians with no formal training in logic. They work hard to help their students, but need a text to support the transition. They count on the text clearly explaining and illustrating new concepts, definitions and the conventions of writing proofs. They emphasize proving theorems and cover elementary set theory. Often individual departments add further requirements, such as introducing discrete mathematics or providing a head start in real analysis or algebra.
One of the best selling texts, A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre and published by Brooks Cole, succeeds in many fundamental ways. It covers all of the core material with a large number of problems of many types. It also highlights the different proof formats. However, my colleagues and I were dissatisfied with it for several reasons. Its encyclopedic presentation of properties in elementary logic and set theory makes it difficult for students to distinguish useful properties from incidental ones. Its dry presentation of mathematics without motivation or connections failed to engage my students—they thought the course was a waiting room before they were allowed to learn the real stuff. Like other texts, it doesn’t convey intuitions about new definitions, notations and concepts, and it expects students to incorporate this new material into proofs immediately.
One of the oldest texts, How to Read and Write Proofs by Solow and published by Wiley, accomplishes well what its title says. In 1983 I used it successfully as a supplement to a linear algebra course emphasizing proofs. (We no longer try to teach proof techniques in this course.) However, it would hardly qualify as the text for a foundations course since it doesn’t cover set theory or other mathematical content.
The book Chapter One by Schumacher and published by Addison Wesley, uses a decidedly different approach from other texts. It intentionally does not provide proofs for the results so that students will need to construct their own text. Unfortunately it is thus not useful as a later reference for students. A colleague used it for a small discussion based section of our foundations course. She thought it worked reasonably well for that format and class size. I suspect that few professors would choose such a course structure. I doubt this text would fit well with a standard course or with more typical class sizes.
I just finished using The Foundations of Higher Mathematics by Fletcher and Patty and published by Brooks Cole. My colleagues advised me that it was the best available, better than the text by Smith, Eggen and St. Andre. While I like much of it, and the students generally find it more readable than Smith’s text, I find many aspects of it frustrating. Indeed, my disappointment with this text has finally pushed me to propose writing a text. It doesn’t emphasize proof formats enough and fails to discuss how to prove uniqueness in the chapter on proofs. It fails to motivate or inter-relate the various topics it presents. The problems are uneven in quality. The material on the axiom of choice and its equivalents is too brief to help students understand how to use these sophisticated tools. The authors too often write definitions and problems using confusing colloquial English. Some of the biographical sketches distort the history, most egregiously the biography of Evariste Galois.
The texts Introduction to Mathematical Structures and Doing Mathematics by Galovich and published by Harcourt Brace and Jovanovich are very good, but the first is too sophisticated for many majors, and the second, leaner one omits too many topics.
I have not had the opportunity to use the very recent text Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni and Zhang and published by Addision Wesley. It does introduce set theory ideas prior to proof formats and appears to give good clear discussions about proofs. However, starting with the chapter on equivalence relations it appears to revert to the usual format of expecting proofs at the same time students are learning concepts. Also, it puts relations in front of functions, even though students are much more familiar with functions. Finally, it does not address the Axiom of Choice at all, a topic I think a foundations text should address carefully.
The Proposed Text
My text will include all the familiar topics in the many current texts. The most important pedagogical difference from other texts will be the order of topics in Part I. (See the Table of Contents below.) Students need to work with new notations and definitions before they can prove properties involving them. Therefore I propose delaying proofs until the second chapter. The first chapter will introduce the language of mathematics—logic, sets, and numbers. It will give students practice using and writing definitions, finding examples and counter-examples, and conjecturing. This approach also provides a greater variety of properties to prove in the following chapter on proofs. The large variety of proof exercises, including “outline proofs,” where students fill in missing steps, will reinforce the interplay of intuition and format. I will end the proof chapter with a general discussion of generating and writing proofs, including good proof writing, common mistakes and the role of examples and illustrations.
I propose postponing the chapter on relations until after the chapter on functions. Although functions are formally a special case of relations, calculus students already think about functions in general, whereas they have encountered only isolated examples of relations. Thus students can build on their understanding of functions to prepare their intuition for the abstract properties of relations.
Student learning centers on the problem sets. I will include a large selection of a variety of problems and levels of difficulty, enough for two assignments on material students find difficult. Indeed, a number of sections, which I will indicate in the introduction, deserve more than one class period. Pedagogically I think students benefit from such intentional extra time to synthesize hard material. I will include exercises in reading mathematics, a skill few texts try to address.
I would expect an instructor using this text to do all of Part I. A semester course could include at least one chapter in Part II or substantial portions of two or more. I will write each chapter in Part II to serve as a useful reference for students whose course did not treat that topic. The whole text will act as a long term reference, with the important definitions and theorems clearly indicated in the text and minor ones placed as exercises.
Other proposed features of this text deserve some mention.
This text will incorporate historical perspective and biographical sketches for their intrinsic interest and for pedagogical reasons. For example, the limitations of Aristotle’s syllogisms help explain the need for quantifiers to handle the subtleties of mathematical reasoning.
Many future graduate students need a reference giving a clear presentation of the Axiom of Choice, Zorn’s Lemma, and the Well Ordering Principle. Chapter 5 will explain them carefully and illustrate their use in proofs, although I will write it so the instructor can easily omit that material when presenting the rest of the chapter.
While axiomatics, metamathematics and philosophy of mathematics are unusual topics in a foundations course, I think they belong in the text, especially in its role as a reference. The axiomatic approach has shaped current presentation and practice in algebra, geometry and analysis. In addition, the results of metamathematics inform our understanding of what mathematics and computer science can and can’t do. Questions about mathematical existence, truth and applicability continue to puzzle anyone interested in mathematics, even if the philosophical battles have long faded. In addition to describing briefly the strengths, weaknesses and appeal of traditional philosophical positions I will include short discussions of pedagogical constructivism and cognitive psychology. My students find this material engaging and a valuable opportunity to reflect on mathematics.
The final three chapters focus on particular needs. Some schools include discrete topics in the foundations course to avoid requiring another course, especially for future secondary mathematics teachers. The chapter on discrete mathematics will fulfill this need. Abstract algebra students’ greatest difficulty is to operate formally in proving abstract properties. The algebra chapter will provide a bridge using familiar examples. Students will make conjectures as well as do simple proofs. Students in analysis often struggle with its sophisticated definitions and proof techniques. As preparation the chapter on analysis will build an analytical intuition about “turning approximations into exact values” before introducing formal definitions and delta-epsilon proofs.
Table of Contents
Foundations of Mathematics
Thomas Q. Sibley
Chapter 1. Language, Logic and Sets
Section 1.1 Logic and Language
Section 1.2 Implication
Section 1.3 Quantifiers and Definitions
Section 1.4 Introduction to Sets
Section 1.5 Introduction to Number Theory
Section 1.6 Additional Set Theory
Definitions from Chapter 1
Algebraic and Order Properties of Number Systems
Chapter 2. Proofs
Section 2.1 Proof Formats I: Direct Proofs
Section 2.2 Proof Formats II: Contrapositive and Contradiction
Section 2.3 Proof Formats III: Existence, Uniqueness, Or
Section 2.4 Proof Format IV: Mathematical Induction
The Fundamental Theorem of Arithmetic
Section 2.5 Further
Advice and Practice in Proving
Chapter 3. Functions
Section 3.1 Definitions, Notation and Examples
Section 3.2 Composition, One-to-one, Onto, and Inverses
Section 3.3 Images and Pre-images of Sets
Definitions from Chapter 3
Chapter 4. Relations
Section 4.1 Relations
Section 4.2 Equivalence Relations
Section 4.3 Partitions and Equivalence Relations
Section 4.4 Partial Orders
Definitions from Chapter 4
Chapter 5. Infinite Sets
Section 5.1 The Size of Sets
Section 5.2 Countable Sets
Section 5.3 Uncountable Sets
Section 5.4 The Axiom of Choice (optional)
Definitions from Chapter 5
Chapter 6. Introduction to Discrete Mathematics
Section 6.1 Graph Theory
Section 6.2 Trees and Algorithms
Section 6.3 Counting Principles I
Section 6.4 Counting Principles II
Definitions from Chapter 6
Chapter 7. Introduction to Algebra
Section 7.1 Operations
Section 7.2 Groups
Groups in Geometry
Section 7.3 Rings and Fields
Section 7.4 Lattices
Section 7.5 Homomorphisms
Definitions from Chapter 7
Chapter 8. Introduction to Analysis
Section 8.1 The Real Numbers, Approximations and Exact Values
Section 8.2 Limits
Section 8.3 Continuous Functions and Counter-Examples
Counter-Examples in Rational Analysis
Section 8.4 Sequences and Series
Section 8.5 Discrete
The Intermediate Value Theorem
Definitions from Chapter 8
Chapter 9 Metamathematics and Philosophy of Mathematics
Section 9.1 Metamathematics.
Section 9.2 Philosophy of Mathematics.
Definitions from Chapter 9
Appendix: The Greek Alphabet
Answers to Selected Problems.