The author says that his goals are to show the reader that some aspects of mathematics do not correspond to everyday experience, that we can modify rules and devise new mathematics, and that new mathematics can describe the physical world. This is no news to members of the MAA but not everyone among the general public, for whom the book is intended, is aware of it.
To achieve his goals he uses the negative numbers, which, he says, “stand as just about the only kind of numbers about which a book like this has not been written”. (He forgets about algebraic numbers, not to mention Mahler’s S-numbers.) The means that he uses is, mainly, to suppose that the product of two negative numbers is negative. Unfortunately, he does not make clear the consequences of this assumption (many familiar properties have to be abandoned, the distributivity of multiplication over addition among them) nor, later on, those of supposing that ii = –1. About that, he writes, “It might perhaps seem that if we establish a new value for the expression [ii] then we throw away a lot of useful mathematics. But this is not necessarily true.” Not necessarily true, perhaps, but true nevertheless.
The author goes on and on about such things as the three kinds of numbers (positive, negative, and signless), and his new operations of distinction and partition. It reads quite a bit like some of the works of crank mathematicians. I fear that the book can do nothing but create confusion among its readers and reinforce the notion that the rules of mathematics are arbitrary and senseless, and hence that mathematics itself shares those properties.
The author, a Lecturer in the Department of History at the University of Texas, is no mathematician. He refers to the “cubic roots” of 8, where we would say “cube roots”, writes that “ has only two solutions”, and uses the symbol “⇔” in a way new to me.
There is nothing here for any member of the MAA.
Underwood Dudley has retired from DePauw University and is now living in Florida.
Chapter 1: Introduction 1
Chapter 2: The Problem 10
Chapter 3: History: Much Ado About Less than Nothing 18
The Search for Evident Meaning 36
Chapter 4: History: Meaningful and Meaningless Expressions 43
Impossible Numbers? 66
Chapter 5: History: Making Radically New Mathematics 80
From Hindsight to Creativity 104
Chapter 6: Math Is Rather Flexible 110
Sometimes -1 Is Greater than Zero 112
Traditional Complications 115
Can Minus Times Minus Be Minus? 131
Unity in Mathematics 166
Chapter 7: Making a Meaningful Math 174
Finding Meaning 175
Designing Numbers and Operations 186
Physical Mathematics? 220
Further Reading 249