Getting to the Point book photo

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:

Goals

Features

Basic prerequisites

Problem-focused

Additional material

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:

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