Mathematicians have a very ambiguous relationship to visual thinking in mathematics. Virtually all of us use visual thinking, if not in our own research, then in our teaching. However, we are also suspicious of it, because it can easily lead us astray, to believing, for example, that a continuous function must be differentiable except at a discrete set of points.
In the last twenty years there has been a resurgence of interest in the role of visual thinking in mathematics, beginning with work of Barwise and Etchemendy in 1991. Giaquinto’s book is an important contribution to this work. Its aim is to show that visualization can lead to mathematical discovery and, in some cases, to mathematical knowledge.
The book discusses a wide range of topics involved in visualization, including visual aspects of calculation (as calculation by hand is actually done), getting general theorems from specific images (such as the famous proof that the sum of the first n integers is n(n+1)/2 using a triangular array of dots), symbol manipulation, cognition of structures such as trees, and why visualization is so unreliable in analysis as compared to geometry.
However the heart of the book is its discussion of geometry in chapters 2 through 5. Although the author discusses the role of diagrams in geometric proofs in chapter 5, the primary focus of the book is really the role of visualization in geometric discovery, chapter 4. The main theses are:
visualization can be a reliable route to discovering new geometric facts (by discovery, he doesn’t necessarily mean facts unknown to the community of mathematicians, but facts unknown to the person discovering them, and found independently, without being told);
such discoveries can be genuine knowledge;
these discoveries do not involve inference from sense experience;
and thus such discoveries are examples of synthetic a priori knowledge.
Chapters 2 and 3 (Chapter 1 is the introduction) set the stage for this. In chapter 2, he describes what is involved in visualizing a geometric object — he focuses, for his example, on squares — and on the concepts one develops that are intimately involved in these visualizations. He first introduces what he calls the “perceptual concept for squares” and then uses this to describe what he calls a “geometrical concept for squares.” Chapter 3 describes how one can obtain what he calls “basic geometric knowledge”, first, by describing how one obtains beliefs about geometric objects, via “belief-forming dispositions” that are not empirical, and why the beliefs so obtained should be counted as knowledge.
The arguments are quite delicate, and people who have no patience for philosophical discussions should steer clear of the book. However, I found most of what he said to be well thought out and fairly reasonable, from a mathematician’s perspective. There are some places where I have substantial concerns about what he says. For example, he chooses to use, as his description of the perceptual concept for squares, having straight edges parallel to the horizontal and vertical axes of a given coordinate system; but he never says that it has exactly 4 sides. His description of a square is satisfied by a regular octagon or any regular 2n-gon. (There are philosophical reasons perhaps for choosing to avoid using numbers. However, he could say, at the end of his description, that the described sides are the only sides of the figure, which would rule out figures with a larger number of sides — but he doesn’t!) In his description of the geometrical concept for squares, he claims that “It can also be part of the content of experience that a square is perfect.” (p. 28) Similarly, on p. 29, he says “possession of an initial geometrical concept of squares centres on a disposition to judge something square just when it appears perfectly square to one and one trusts the experience.” But one knows (and he acknowledges, later in the book) that one has no way to determine if a physical object, no matter how it appears, is perfectly square — so how can one trust the experience of it appearing so? He also, suddenly, in the transition from chapter 2 to chapter 3 changes terminology from “geometrical concept of square” or simply “square” to “perfect square”, which I found disconcerting. However, these appear to me to be minor matters that should be fairly easy to fix.
Giaquinto does an excellent job of distinguishing situations where visualization can lead us astray, via unwarranted generalizations or features of a diagram that are not common to all members of the class under consideration, and proposes a very plausible explanation of why visualization in analysis tends to lead one astray more often than in geometry.
As an added bonus, the book is a contribution to a platonic view of mathematics. Kurt Gödel famously asserted that we form our ideas of mathematical objects on the basis of something else which is objective and immediately given, but not, or not primarily, the sensations. These data may “represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality.” (Gödel, p. 484, “What is Cantor’s Continuum Problem”, in Benecerraf and Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge University Press) As I see it (and the author may well disagree completely), Giaquinto’s work makes an important contribution to describing this other kind of relationship that Gödel is referring to, and that most philosophers treat as simply mysterious and therefore illegitimate. Platonism is viewed by many as not viable because, if mathematical objects are abstract, there appears to be no way that physical beings can gain knowledge of them. Giaquinto’s work gives at least a partial response to this objection.
The book is relatively free of philosophical jargon, and should be of considerable interest to anyone interested in the philosophical discussion of the wide range of ways we use visualization on our mathematical reasoning.
Bonnie Gold is in the Mathematics Department at Monmouth University. Her Ph.D. was in mathematical logic, but in recent years her work has been in undergraduate mathematics education (especially assessment: she’s co-chair of the MAA’s Committee on Assessment) and in the philosophy of mathematics. She recently co-edited (with Roger Simons) Proof and other Dilemmas: Mathematics and Philosophy, published by the MAA in 2008. She is a co-founder (first chair, and currently Public Information Officer) of the SIGMAA on the Philosophy of Mathematics. She’s also a nut about Japanese gardens, which tax her minimal visualization skills.