On the Study and Difficulties of Mathematics

Most MAA reviews are written about new books; the book reviewed here, reprinted by Dover as a Phoenix Edition in hardcover, comes to us from almost two centuries back. Its author Augustus De Morgan (English algebraist and logician, today best known among mathematicians for the De Morgan Laws about unions and intersections which he formalized) is long gone and will never read this review. Obviously I don’t need to hold any punches. You might expect this review will therefore be blunt, and tell you the good, the bad and the ugly.

Let us start at the beginning with a brief look at what the book is about. On the Study and Difficulties of Mathematics is intended to be a companion to students of elementary mathematics venturing out into their mathematical journey alone, without “the advantage of a tutor” (p.1). Its main goal is to describe the fundamental difficulties such students may encounter, and resolve them by providing clear and explicit explanations. In particular, De Morgan is not much interested in the philosophical foundational questions that may intrigue the more advanced student, but in the doubts and confusions that haunt the innocent. By providing a study guide of sorts, he intends to show the student a clear path through the main pillars of arithmetic, algebra, geometry and trigonometry.

There are a few reasons why readers of this review may decide to pass this book by. First of all students of today may prefer a more user-friendly and colorful text with shorter sentences and smaller words. The book’s language and style reflect its times, and its author assumes that the reader is capable of handling elegant but lengthy sentences about mathematical ideas and constructions. There are some figures and illustrations, but many pages go in between with none.

Perhaps more significantly, some of the material is hopelessly (out)dated. Mathematics has changed tremendously since the 1830s, when this book was first printed. While De Morgan was trying hard to find a sensible interpretation for negative numbers and imaginary numbers, today’s mathematician handles them without thinking and is completely at ease in their presence. While geometry in De Morgan’s time was exclusively Euclidean and represented the truth that our senses presented to us, today’s mathematician has to be more cynical, knowing the real reason why it has been so difficult to prove the fifth parallel postulate. While De Morgan spent a whole section on logarithm tables, even the most luddite of today’s mathematicians has abandoned them and given in to the lure of the calculator.

There is much to be savored here, however. De Morgan’s personal style shines through the pages. He does not pull any punches himself when he explains why he wrote the book:

The number of mathematical students, increased as it has been of late years, would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present. If any one claiming that title should think my attempt obscure or erroneous, he must share the blame with me, since it is through his neglect that I have been enabled to avail myself of an opportunity to perform a task which I would gladly have seen confided to more skilful hands. (preface, viii)

We see and sympathize with his lofty goal of helping the student who has been or will otherwise be left behind. De Morgan is not the stereotypical research mathematician who cannot be bothered with the woes of the common student.

De Morgan defends the place of mathematics in a liberal education in his initial chapter. He first asserts its intrinsic difference from other paths of study; in his own words “the nature of mathematical demonstration is totally different from all the other, and the difference consists in this that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible.” (p.4) In this chapter already the modern reader may notice the dated outlook that sees (Euclidean) geometry as a presentation of exact truths about our world; however, the dominant theme of the distinct nature of mathematical demonstration is still convincing even if we disregard this Euclidean bias.

According to De Morgan the study of mathematics is good for developing one’s reasoning skills. We all still say this, but he actually attempts to justify his claim. In particular he lists five points that, according to him, distinguish mathematics from other disciplines and make it most suitable to be used as training ground for reasoning skills. Among these the specificity of mathematical definitions and the emphasis on logical deductions are perfectly acceptable points to raise, while Euclidean geometry and its connection to reality come into play in the remaining ones. As we are part of his choir, however, and most likely agree about the importance of mathematics in education, we can perhaps slip some of that under the rug.

Mathematical instruction begins with the second chapter. We start from the basics, the natural numbers and the symbols that stand for them. We learn about base ten, and how to use the symbols that represent the four arithmetic operations. We then move on to basic rules of arithmetic, and then to fractions and numbers with decimal expansions. Along the way De Morgan points out that some fractions cannot be represented as numbers with (finite) decimal expansions but we can approximate these with such as close as we like. Here the modern reader is thinking of the real numbers while the typical advanced student is probably thinking of limits. De Morgan simply mentions approximations, and in there is hidden a whole world of elementary analysis.

This is a typical theme in the rest of the book. De Morgan is always careful with his words, and even though he never goes into very advanced material, the modern mathematician can fill in the details to make the connections. Even the negative numbers which De Morgan so distrusts (For example, on p. 72: “It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing […] above all, that the notion should have outlived the belief in judicial astrology and the existence of witches, either of which is ten thousand times more possible.”) are explained eventually in a manner which can be interpreted accurately within today’s mathematics via the use of the two directions of the number line and vector quantities.

On a different note, proponents of active learning may enjoy finding out that De Morgan was one of them. At some point he does assert that “if [the reader] can read and understand all that is set before him, the essential benefit derived from mathematical studies will be gained, even though he should never make one step for himself in the solution of any problem” (p.96). He argues explicitly, however, for letting the students discover the rules of algebra for themselves:

The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connexion with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given, of which the details are suppressed […] These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand while engaged in any algebraical process. (pp.175–176).

In several other instances he emphasizes that rules should not be accepted as facts before the student feels comfortable with what they mean in specific instances and understands them fully.

Along the way, De Morgan makes some comments about the nature of our mathematical knowledge and how we extend it. While discussing the arithmetic operations of fractions, and then later on when describing algebraic operations, he refers to two comprehensive principles. The first is that in mathematics pedagogy, the soundest path is to begin with the special cases and specific examples and then move on to the general theory: “It is by collecting facts and principles, one by one, and thus only, that we arrive at what are called general notions” (p.33). A second principle that he often refers to involves extending rules and scopes of various terms as we discover their relevance and usefulness in these broader contexts: “suppose that when we have discovered and applied a rule and given the process which it teaches a particular name, we find that this process is only a part of one more general, which applies to all cases contained under the first, and to others besides. We have only the alternative of inventing a new name, or of extending the meaning of the former one so as to merge the particular process in the more general one of which it is a part.” (pp.33–34). Making these principles explicit, De Morgan aims to help his readers avoid unnecessary confusion which may otherwise arise.

Reading De Morgan’s On the Study and Difficulties of Mathematics, I found myself thinking critically about elementary mathematics, material that I used to take for granted. Furthermore I kept noticing how much more we know today and what we easily accept as the correct way to think about mathematics. I could not help but wonder about what in our approach to mathematics today will feel outdated to mathematicians of the future. Besides being a historical witness of its times, the book taught me some excellent ways of approaching elementary mathematics, and further extended my understanding of its connections to more advanced matters. I understand why in 1917 the MAA Library committee recommended it among the 160 books “suitable for purchase by the usual school or college library”. (“A List of Mathematical Books for Schools and Colleges”, The American Mathematical Monthly, Vol. 24, No. 8 (Oct., 1917), pp. 368–376.) Both as a historical artifact and as a thoughtful treatise on elementary mathematics that still can make us think, De Morgan’s book has kept its value.

Gizem Karaali is assistant professor of mathematics at Pomona College.