Getting to the Point

This is the website for the forthcoming mathematics textbook “Getting to the Point,” by Daniel Goroff and Nat Kuhn, a compelling, careful, unorthodox, and purposeful introduction to higher mathematics.

Our draft manuscript has been getting positive reviews, including from a Fall 2021 pilot at UMass Boston, where more than one student said it was “the only math book I’ve ever enjoyed reading.” We are particularly interested in folks who might want to pilot the book for Fall 2022. If you or someone you know might be interested in piloting or reviewing the draft, please contact Dan (goroff AT sloan DOT org) or Nat (nk AT natkuhn DOT com).

For classes interested in piloting the book, we have been able to provide printed copies on fairly short notice at $15 each.

Rationale and History

The sharp divide between lower-level and upper-level mathematics courses is in students’ ability to write and appreciate mathematical proofs. As a result, many colleges and universities have courses to teach students basic proof-writing skills.

When Dan taught a course of this sort, he was not satisfied with existing texts. In many of those “bridge course” books, nothing seemed wrong—but nothing seemed particularly coherent or complete either. Some were like reading a cookbook with chapters on how to use a knife, how to use an electric mixer, and how to measure ingredients, without any direct discussion of how to cook meals that are delicious and satisfying. In addition, many of the books seemed to skirt or evade questions of mathematical foundations while at the same time insisting on the need for total rigor.

This book is an expanded and reworked version of the notes Dan developed, with assistance from Michael Hutchings, as an alternative. Classes based on those notes have been successfully taught to:

• undergraduates interested in mathematical fields at a research university;
• math majors at a state university where few go on to graduate school;
• Ph.D. candidates in fields other than mathematics;
• summer programs for high school students;
• future teachers of K-12 mathematics; and,
• math professors wanting to experience something new, through NSF’s Chautauqua Program.

Goals

• The overall goal is to introduce students to the knowledge, skills, and attitudes that they will need in any upper-level study of mathematics, by providing an inclusive invitation to what sets advanced mathematics apart.

• Rather than presenting a smattering of techniques and results, Part I of the book starts with an informal description of the “crumpled paper theorem”—the Brouwer fixed-point theorem (BFPT) in two dimensions—and proceeds to build up all the machinery necessary to state and prove it rigorously. The BFPT is a substantial, non-obvious mathematical result that is usually considered too “advanced” to for students at this level.

• What is a proof? The book starts with the heuristic that a proof is a compelling argument: one that can withstand all challenges. Written proofs will vary greatly in the amount of detail presented, but if you are presenting a proof you need to be ready, in principle, to defend any step and respond to any question about it.

• There is a lot to unpack in order to understand and prove the BFPT: What is a theorem? What are axioms and how do we use them? Over the course of Part I, students are exposed to a broad variety of useful definitions, constructions, and techniques, with particular emphasis on the fundamentals of point-set topology. We introduce a number of definitions in non-standard but carefully motivated ways. As a result, the proofs of even familiar results can look surprising and thought-provoking.

• From the very first pages, the book is motivated by the following challenge: Suppose you have a friend who is very skeptical, thorough, and imaginative, and who asks lots and lots of questions accordingly. What would it take to convince this person that Brouwer’s claim about the existence of a fixed point is really a theorem?

• While adhering to and instilling a high standard of rigor, the book takes a relatively conversational tone, introducing students to a wide range of styles of mathematical discourse.

Features

Basic prerequisites

• Experience and comfort with high school algebra is necessary. Some previous exposure to calculus is helpful but not necessary. But even for students with much more substantial backgrounds, the material here will likely seem orthogonal and fresh.

  Get students doing “real mathematics” as soon as possible

• The focus of Part I is a proof of the two-dimensional Brouwer fixed-point theorem, a substantial result which is usually not accessible at this level.
• In order to not get bogged down with excessive prerequisites, the “background” material is presented in Chapters 1, 3, 4, 6, and 8, with the beginnings of point-set topology carefully interwoven in Chapters 2, 5, and 7.
• There is quite a bit more explanation and hand-holding at the beginning; as the book progresses, students should develop the ability to fill in more details on their own.
• As originally taught, the course presented prerequisites and techniques in a “just-in-time” fashion, as they were being used. If the book were structured that way, the prerequisites would have more widely scattered, making the material difficult for students to locate. However, it’s possible to teach it in a more “just-in-time” way than following the text section-by-section.

Problem-focused

• In order to involve students in active learning, many exercises give students an opportunity to review and complete results, with significant structured hints. In addition to giving more practice opportunities, it also breaks proofs up into smaller pieces, so that students’ eyes glaze over less.
• There are some good “PDSP” problems (“prove or disprove and salvage if possible”), with more to come.
• Some entire topics are presented as structured exercises, such as the optional Section 10.6 on connectedness.
• There are problems going in various directions for projects, or for students who are interested in doing more advanced work. For example, differential calculus is developed as far as Taylor’s theorem with the Lagrange remainder.

Additional material

• There is much more material than could be covered in a single semester.
  • Part I—going from zero to a proof of the two-dimensional BFPT—is designed to be covered in a one-semester course, which is likely to be the main classroom use for the book.
  • Part II covers a careful proof of the $n$-dimensional Brouwer theorem, including all the necessary linear algebra and affine geometry.
  • Part III contains all relevant foundational material in logic, axiomatic set theory, and construction of number systems. There is a treatment of formal logic including a proof of Gödel’s completeness theorem, and a discussion of Gödel’s incompleteness theorems with an outline of the proofs.
  • The book is designed as a natural segue into project- and inquiry-based learning, with extensive optional exercises and sections devoted to additional material.
• Depending on the level of the students, material can be omitted from Part I, or added.
• There are many possible uses for the material:
  • The book could easily be used for a full-year course in mathematical foundations and point-set topology, similar to many introductory courses in real analysis.
  • After a semester covering Part I, an optional second semester could cover more advanced topics in a project/inquiry-based format.
  • Some of the additional material could be used by individual students who are ready for more challenge than the class as a whole.
  • We anticipate that some students will want to hold on to the book and refer to some of the additional material on their own, particularly the sections on foundations.
  • The book can be used for self-study, for bright high-school students on up. Working through it with a partner or in a small group would be even better.

Honesty and rigor

Since the philosophy of the book is that students should be prepared to defend and justify every step of their proofs, it seemed important that our text should live up to the same standard. This is why the foundational material is covered in such detail in Part III.

Off-the-beaten-path approach

When our definitions diverge from the standard, we work to be clear about what we are doing connects to the usual approach. The most prominent example is that, rather than starting with $\mathbb{R}^n$ or metric spaces, we start with topological spaces, defined via Kuratowski’s axioms for closure operators. This has the following advantages:

• It is a gentle introduction to reasoning carefully from axioms.
• Students don’t have to think about “the set of all open sets” right at the start, which is too abstract a starting point for students at this level.
• The definition of continuity is very intuitive. Once you get used to it, it’s actually very nice to work with.
• There is a single framework that covers both continuous functions and limits of sequences; in addition to being convenient, it’s a good example of the power of abstraction.
• It will be harder to find answers on the web (at least for now).
• It’s good for students to understand that there are multiple approaches to things, and that that’s fine as long as you can show that the various approaches are equivalent.

Contents

The table of contents does not really give an adequate sense of the flavor of the book, but with that caveat, here it is.

About the authors

• Daniel Goroff earned his Ph.D. in mathematics at Princeton and also holds graduate degrees in economics and in mathematical finance. He taught at Harvard for over twenty years before becoming Dean of the Faculty and Vice President for Academic Affairs at Harvey Mudd College. His service in Washington includes leadership positions at the National Academy of Science, the White House Office of Science and Technology Policy, and the National Science Foundation. He is currently Vice President and Program Director at the Alfred P. Sloan Foundation.

• Nat Kuhn received a Ph.D. in Mathematics from Princeton under the late William Thurston. He was a Putnam Fellow and C.L.E. Moore Instructor in Mathematics at M.I.T. Now a psychiatrist and psychotherapist in the the Boston area, he has written several books on psychotherapy and teaches and supervises therapists internationally.