Has it ever bothered you that many mathematics textbooks begin with a number of strangely crafted definitions?
These definitions invariably turn out to be extremely valuable for the development of the theory in question, but it may be days, months, or years—with many rereadings—before this is apparent.
I remember wondering as a college sophomore why the definition of the linear independence of vectors v1, v2, …, vn had to be so complicated and, in particular, what the algebraic condition
The only scalars a1, a2, …, an that satisfy a1v1 + a2v2 + … + anvn = 0 are a1 = a2 = … = an = 0.
had to do with the ability of the vectors to fill out n-dimensional space. It was years before I figured out that connection.
Anyway, after the cryptic definitions, the textbook author embarks on proving a series of theorems whose purpose is hidden until quite late in the theory’s development, if ever.
The most important results, so identified because they are called The Fundamental Theorem of Something or Other, appear at the end of section 3.3 as a corollary to some other impenetrably technical theorem, apparently as an afterthought!
If the intrepid reader has lasted this far, the author throws a bone late in chapter 4 in the guise of an application of the theory to some problem that may have helped someone at some time.
No wonder so many people think that mathematics is only for nonhumans!
To be fair, more and more mathematics textbooks are far better written than this caricature I have painted for you, but sadly, plenty of examples of expositional writing in mathematics fit this mold.
You can thank Euclid for this penchant professional mathematicians have of organizing their writing in axiomatic form: definitions, axioms, propositions, theorems, and corollaries. Indeed, Euclid didn’t bother adding applications to such expositions. You can thank Archimedes, Ptolemy, and Galileo for including them (although Descartes, Gauss, and Cauchy usually did not).
I won’t argue that the traditional axiomatic style lacks value—mostly because I don’t believe this at all! I will suggest that it is not the best vehicle for learning mathematics. My contention here is that the best antidote for students who struggle with traditional forms of axiomatic exposition is to investigate the history of the subject.
That definition of linear independence at the start of this article? I finally started figuring out the link between it and the geometry of space when I read about linear algebra’s history (specifically the work of Herman Grassmann in the mid-1800s and the later formalism of linear algebra under Giuseppe Peano and others later in that century). And this is not the only occasion when learning how a mathematical subject developed helped me make sense of what was going on.
Studying the history of mathematics has much to offer the mathematics student:
- Context (the conceptual and cultural circumstances for the underlying problems);
- Motivation (the rationale or even the value of wanting to know the answers to the central questions involved); and
- Connections (how the mathematicians thought in terms of other ideas that were already established at those times and places).
To purchase at JSTOR: http://dx.doi.org/10.4169/mathhorizons.22.4.34
Danny Otero is an associate professor of mathematics at Xavier University, president of the MAA Ohio Section, and chair of the History of Mathematics Special Interest Group of the MAA. He still digs the Power Puff Girls. Email: firstname.lastname@example.org