Numbers are unusual. Numbers
escape ordinary apprehension since we cannot perceive them directly using one
or more of our five senses. By numbers, we usually think of whole numbers, i.e.
integers, such as 1, 2, or 3 or fractions such as a half, a quarter, or three
quarters. Integers and fractions are real numbers. The notion of a real number
suggests two things: (1) that we can know these numbers exist and (2) that
there are other unreal or imaginative numbers. 
The latter are inexpressible as a fraction or whole number, which
implies that its decimal expansion goes on forever without repeating itself.
Some examples of imaginary numbers include e, pi, or the square
root of -1. In Imagining Numbers, Barry Mazur explores the issue of
imaginary numbers and whether we can conceive of them as truly existing
objects.

Mazur has divided Imagining
Numbers into three parts. The first part outlines Mazur’s thoughts about
the relationship between imagination and numbers. Mazur argues that numbers are
the imaginings of our intellect. Since we cannot experience a number, numbers
must be discovered using our imagination. We can imagine the number 1 or number
4,007. This is not to say that the number exists in our mind; rather, using
reason is the only way we become familiar with numbers. According to Mazur,
when we think of a number, we have a mental image of the number in our minds.
Imagining imaginary numbers is a bit more complex than imagining real numbers.
Mazur claims that we need to know how to apply simple rules of arithmetic to understand
complex numbers, such as the square root of -15. The remainder of the first
part is a mathematical demonstration of how to understand the simple laws of
arithmetic, like the distributive law and why a negative number times a
negative number yields a positive, to show that understanding what complex
numbers are is possible. Mazur’s argument, however, never indicates what a
number is even though he claims we can use reason to imagine a number.

The second part of the book
furthers the analysis begun in the first part. This section of the text shows
how the simple laws of arithmetic apply to complex and unreal numbers. Since
Mazur has failed in the first part to tell us what real numbers are, he leaves
the same stone unturned in his analysis of unreal numbers. We are supposed to
assume that if we understand how to apply the laws of arithmetic to numbers,
real or unreal, then we know what numbers actually are. Mazur believes the
application of laws reveals the ontological status of imaginary numbers. The
argument is unconvincing. Applying laws to some object is unrelated to whether
the object exists. If an object exists, then it may be susceptible to certain
laws of physics. There are some objects, however, that are not subject to
natural laws, e.g. black holes, and some natural laws may not apply to objects.
It seems reasonable to remain skeptical about whether applying laws of
arithmetic bares the ontological status of numbers.

The final part relates to the
literature of understanding and discovery. Mazur asserts that no formal
analysis of the words yellow and tulip will capture the object we imagine.
Mathematics seems quite the opposite. We are able to formally analyze the
concepts of mathematics, but it seems we cannot have sensations of mathematical
concepts. Mathematical truths are magnificently enclosed within their formal
properties and formal consequences. In the same imperfect way that a formal
analysis of the words yellow and tulip fail to reveal the sensations we
encounter when we perceive a yellow tulip, according to Mazur, mathematics
turns out to be imperfect because we can formally analyze mathematical concepts
without being able to empirically perceive such concepts.

The aim of the book is to show
that imagining unreal numbers is analogous to generating a mental image of an
object from the words we see written on a page. Mazur uses the notion of the
yellow of the tulip to show that just as we can imagine a yellow tulip from the
words we see on a page, imagining unreal numbers is very easy so long as we
correctly understand the relevant mathematical rules to form ideas about such
numbers. What Mazur seems to overlook is that there is an equally strong
disanalogy with the yellow of the tulip example. We, at some future time, will
be able to apprehend the yellow tulip using ordinary sense perception. Suppose
that we visit the Netherlands where there are many yellow tulips. We will be
able to perceive many yellow tulips. We cannot experience numbers, even real
ones, in the same way. We cannot travel to Cambridge, Massachusetts to frolic
in a field of numbers. To say that imaginary numbers are easily apprehended in
the intellect because we can have a sense of the yellow of the tulip without
having one immediately present to experience is ignoring the fact that we have
never had or been able to experience a number. Despite this shortcoming of
Mazur’s book, it is an imaginative and interesting portrayal of mathematical
concepts. Also, the accessibility of Mazur’s prose makes Imagining Numbers
an entertaining read for a general or scholarly audience.

©
2003 Joseph W. Ulatowski

Joseph W. Ulatowski, Department of
Philosophy, University of Utah

Categories: Philosophical, General