That article by Lockhart (referred to in a previous post of mine) was linked from Keith Devlin’s page at the MAA site. I ended up surfing through his previous articles. There is a good pairing on what mathematics education should be, not at the elite level (preparing future mathematicians) but the general level (ensuring that we all have basic literacy). The main outline can be found here, but it refers to an earlier article here. That article led me to Devlin’s course at Stanford (outline and philosophy are both worth reading and can be downloaded from that link). Which, in turn, led me to his book, The Language of Mathematics. As luck would have it, it was in my public library.
As you can see it has taken me a couple of months to read it but I finished it last night. It was hard going sometimes and I can’t claim to have understood everything, but I think that I now have a better sense of what mathematics is about than I did before. And some areas were truly fascinating and really advanced my own knowledge.
A couple of summers ago, I did some work on the nature of research in the humanities disciplines. It was very interesting. And I have felt for a long time, though without a lot of knowledge, that mathematics is really more like the humanities than like the sciences. Although mathematics is very useful to scientists, mathematicians don’t really approach the world in the same way that scientists do. They are more like philosophers. Reading The Language of Mathematics confirmed this view for me. Mathematicians are a lot like philosophers. Reality does not concern them very much. Abstraction is very important to them. And finding abstract patterns really excites them. Mathematicians put a lot of value on elegance. And simplicity.
An illustration of the importance of beauty and aesthetics to mathematicians can be found in chapter 3 of Devlin’s book. Don’t worry if you don’t quite understand what he is talking about her, I’m not sure I do completely either. My point in quoting it is to highlight the use of aesthetics as an argument for the acceptance of something.
With the gradual increase in the use of complex numbers spurred on by the obvious power of the fundamental theorem of algebra and the elegance of Euler’s formula, complex numbers began their path toward acceptance as bona fide numbers. That finally occurred in the middle of the nineteenth century, when Cauchy and others started to extend the methods of the differential and integral calculus to include the complex numbers. Their theory of differentiation and integration of complex functions turned out to be so elegant — far more than in the real-number case — that on aesthetic grounds alone, it was, finally, impossible to resist any longer the admission of the complex numbers as fully paid-up members of the mathematical club. Provided it is correct, mathematicians never turn their backs on beautiful mathematics, even if it flies in the face of all their past experience. (page 135)
Throughout, Devlin is quite clear that mathematicians are not directly concerned with reality. In the last chapter he spells this out quite clearly when talking about the nature of light.
In what sense is this rapidly moving entity a wave? Strictly speaking, what the mathematics gives you is simply a mathematical function — a solution to Maxwell’s equations. It is, however, the same kind of function that arises when you study, say, wave motion in a gas or a liquid. Thus, it makes perfect mathematical sense to refer to it as a wave. But remember, when we are working with Maxwell’s equations, we are working in a Galilean mathematical world of our own creation. The relationships between the different mathematical entities in our equations will (if we have set things up correctly) correspond extremely well to the corresponding features of the real-world phenomenon we are trying to study. Thus, our mathematics will give us what might turn out to be an extremely useful description — but it will not provide us with a true explanation. (Devlin, page 311-312; emphasis mine)
In some ways this is quite freeing. What we are being introduced to in this book is a new way of thinking; a way of viewing the world through mathematicians eyes. At times this kind of makes your head hurt a bit. And if you really wanted to understand everything he was saying, you would get frustrated very quickly. But if you are willing to let some of it flow over you a bit in an attempt to grasp something of that way of seeing, then it is well worth a read. I suspect that if one was really interested, this book would take several readings. A bit like philosophy really.
Indeed, as noted above, I discovered The Language of Mathematics when investigating Keith Devlin and what he did. He uses this book as the core text for a course he teaches. And it might be the kind of thing that is better taken slowly, over 3 months, pondering each chapter and engaging in actual mathematical work as a means to develop understanding. I’m not sure how easy that would be to do without the guidance of a professor, the structure provided by a course, and the opportunity for discussion in a seminar. But I’m sure it would be worthwhile.
However, for those of us who do not have the time or energy right now to take on that kid of commitment, The Language of Mathematics is still useful book. Individual chapters would be worth reading to get a sense of the bigger picture in some specific field of mathematics. The Prologue gives an overview of the discipline. Chapter 1 “Why Numbers Count”, opens our minds to new ways of thinking of numbers and what we can do with them. Chapter 2 ” Patterns of the Mind” introduces us to mathematical logic and proof, the “discipline” of mathematics not in the sense of “a branch of knowledge” but in the sense of “the practice of training people to obey rules or a code of behaviour” (OED). (These two senses of the term are not unrelated.)
Chapter 3 “Mathematics of Motion” gave me a much better sense of the calculus and what it is trying to do. It is interesting that this topic comes so early in the book and perhaps that has had some influence on my plan to introduce it earlier in Tigger’s education than is usually the case in the school scope and sequence. It is followed by “Mathematics gets into Shape” (Chapter 4) a fascinating discussion of geometry that begins with Euclid and moves on to demonstrate why Euclidean geometry, though terribly useful in everyday life, is not actually the geometry of the “real world”. The 3 angles of a triangle only add up to precisely 180 degrees in the imaginary Euclidean plane. We live on a sphere. For most everyday purposes, even of scientists, the piece of the sphere we live on is so close to a plane that we can ignore this fact and work as if it were in fact a plane. But if we were flying an airplane, the difference makes a difference.
Now if you are the kind of person who likes things to be True and can’t see the point of studying anything that is only approaching the truth or approximating the truth, this might be terribly disturbing. For someone like me who has long embraced a sort of epistemological uncertainty, this just brought mathematics and the sciences into the same set of discussions that the humanities and social sciences have been grappling with in a different room. That’s a lot of big words, sorry. “Epistemology” is just the study of how we know what we know. So the central issue here is not “is there a Truth?” (big T) but “can we know it?”. I have become less concerned with the Truth, and more at ease with approaching it, approximating it, and generally learning. This book takes very much that approach, as the quotes above demonstrate.
Back to the chapters… “The Mathematics of Beauty” (Chapter 5) deals with symmetry, tiling the plane, sphere packing and related issues, introducing the mathematical concept of the “group”. Chapter 6, “What Happens when Mathematics Gets into Position”, introduces topology beginning with the very useful, but not at all to scale, London Underground Map and leading on to the concept of “networks” and some very funky ideas about n-dimensional universes. (I did warn you that mathematicians are not concerned with reality.) “How Mathematicians Figure the Odds” (Chapter 7) is obviously about statistics but introduces some interesting connections among gambling, insurance, and global finance.
Devlin closes with a discussion of light and the universe, “Uncovering the Hidden Patterns of the Universe”, that includes a very simple explanation of Einstein’s theories of relativity that makes perfect sense. (Or maybe it only makes perfect sense once you have read the whole book and come to terms with epistemological uncertainty.) Approaching the problem of the nature of light from a different angle, Einstein developed his theory of special relativity
Building on Lorentz’s theory, Einstein went one significant step further. He abandoned the idea of a stationary ether altogether, and simply declared that all motion is relative. According to Einstein, there is no preferred frame of reference.
The theory of general relativity is an extension of this. I think now is a good time to go back and look at the Einstein exhibition the American Museum of Natural History put together (which I saw when it toured here). As I recall there are some very good illustrations of this principle.
Of course that gets mathematicians back into n-dimensional universes, and the shape of them. That whole curvature of space-time notion is a bit weird from our place on a sphere that looks very like a plane most of the time. But at the end of a book which has thoroughly demolished any idea you might have had that mathematics is some sensible discipline about the real world, it is quite fascinating. Which is the point, I think. Utility, especially immediate utility, limits the pursuit of knowledge so much. Many of the weirder things that Devlin talks about do eventually have some utility but sometimes it takes hundreds of years and several mathematicians to develop these ideas to a point where they can relate to some problem in the “real world”. That isn’t what motivates them to keep working on the ideas though. Mostly, mathematicians keep working on these problems because they are fascinating.
And this brings me back to where I started, wondering about how we teach mathematics. Reading Devlin has opened my mind to a different way of seeing the whole subject. And has confirmed that the objective of mathematics teaching should be to fascinate our students. We must also discipline them: train them to obey the rules or code of behaviour of mathematics as a branch of knowledge. But part of that is just giving them the tools to understand what mathematicians are doing.
And thus even if you can’t face reading The Language of Mathematics, I would strongly urge you to read Devlin’s thoughts on the goals of a mathematics education. Many of them seem perfectly in line with the kinds of things that homeschoolers discuss. In fact, his first goal, the one he says is least common, seems to be a primary goal of the Living Math crowd.
By the way, we should be teaching our children that the rules and codes of behaviour are different in different disciplines. That’s what distinguishes the disciplines as branches of knowledge. The point of learning how to write an elegant mathematical proof is not that this is a “transferable skill” but that this is the way that mathematicians go about demonstrating the truth of something. Although some of the logical principles are also used in other disicplines, practitioners of those disciplines will use them differently, being concerned with a different kind of elegance, perhaps.