What Mathematics is Not
Defining what mathematics is turns out to be surprisingly difficult: despite thousands of years of argument, it is far from a settled issue. Indeed, the journey described on this website is, in a slightly round-about way, my contribution to the debate, describing my own views of what mathematics is. A simpler problem, however, is noting what mathematics is not. This turns out to be a surprisingly significant topic, largely due to apparent confusion in popular culture, and in particular in education, as to what reasonably constitutes mathematics. My aim here, then, is to describe those things that seem to be popularly confused with mathematics, but are, in fact, not.
The particular confusion I wish to dispel is the conflation of mathematics with “facts about mathematics”, or more importantly, the belief that the former consists entirely of the latter. An example of this can be seen in terms of the following comparison of mathematics problems that appeared on the BBC website. The first problem, from China, involves proving the orthogonality and finding angles of certain diagonal lines and planes in a skew quadrilateral prism; the second problem, from England, asks students to find lengths and tangents of angles in a standard 3,4,5 triangle. The point here is that I have had several people tell me that these problems (and other related problems) are testing the same thing and are essentially equivalent — that the Chinese problem just “looks” harder, but isn’t actually testing anything different that is of importance in math. I’m quite adamant that this isn’t the case. Certainly it is true that solving both problems requires the same knowledge of facts about math: in particular the Pythagorean theorem, and the definitions of sine, cosine, and tangent in terms of sides of right angled triangles. You mostly don’t need to know any more facts to be able to solve the Chinese problem. What I feel is lacking from the English problem, however, is the actual mathematics. While the Chinese problem requires you to take the abstract ideas and mentally manipulate them, the English question just requires you to know the facts; no abstract thinking is required for the English question. While the Chinese question requires you to put together a chain of logical argument, and formalise that in writing, the English question requires only blind regurgitation of facts. In short, the Chinese question requires you to actually do mathematics, while the English question only requires you to know facts about mathematics.
At this point some analogies might be in order. A good comparison might be to how history is taught, since it often has similar issues, but the deficiencies are more widely recognised. The sort of difference I am talking about is the difference between presenting history as simply a set of names, dates, and events that need to be memorised, and presenting history as a a matter of analyzing past events and people in context and discussing the significance and meaning of those events. The former case, just names and dates, is simply teaching facts about history, while ultimately history is actually about the meaning of those facts from a historical perspective. In short, history is learning from past events, not just learning learning facts about those past events. Those who fail to learn from history are, of course, doomed to repeat it, as the saying goes.
Another comparison might be made to education in “information technology”. Any geek worth his or her salt will tell you that what matters is learning the underlying concepts rather than particular applications. That is, rather than learning how to use Microsoft Excel in particular, and memorising which buttons and menu entries to click, it is far more profitable to learn the principles of how spreadsheets work. Likewise, it is far better to learn the general ideas of word processors as a whole, such as bold text, headings, bulleted lists, tables, and so on, than specific “click here, now there” recipes for how to achieve particular results in Microsoft Word. Learning the general ideas makes your skills more portable, allowing you to use other software applications, or even newer versions of the applications you learned on. More importantly, by understanding the broad ideas rather than memorising specific recipes, you are far more capable of coping with new problems that differ from previous experience. This, again, is the difference between learning the underlying ideas rather than particular facts.
In both of these examples, despite a certain amount of confusion, and poor teaching as a result, there is general recognition of the difference between the subject proper and just facts about the subject; for mathematics such recognition is far less widespread. For a great many people the facts of mathematics are the be all and end all of the subject. This stems partly from the reality that many facts do need to be learned to make sense of mathematics: you won’t go far without learning your times tables (or some equivalent, such as the Trachtenberg system). The problem is that that is simply where the subject begins. More than once I’ve been told of people’s frustration at having to learn (which is to say memorise) the quadratic formula (or a similar fact or formula) which they are never going to use in ever again after leaving high school. In practice, however, this is little different from having to learn that Columbus sailed to the Americas in 1492. Both are facts that it may be useful to learn to help broaden your understanding of the subject, but learning the quadratic formula is not more important, and no more the “point” of high school algebra, than learning the particular date of Columbus’ voyage is the “point” of high school history. What matters, in the case of a history course, is the context of the voyage, and the subsequent impacts on people, both in Europe and the Americas; learning the date of the voyage is a useful fact that helps provide that context. What should matter, in the case of an algebra course, is learning how algebraic manipulations allow you to reason, carefully and logically, about numbers as a whole, or with unknown quantities; the quadratic formula is a demonstration of how rich and complex that reasoning can be.
Another symptom of mistaking facts about a subject for the subject itself is the so called “mile wide and inch deep” syllabus. When you reduce a subject to just a collection of facts and rote recipes you have the dilemma of which particular facts need to be included. If you are teaching someone how to use a word processor in general, and what sorts of features to look for, you can expect the person to be able to work out how to handle more obscure cases on their own. If, however, you are teaching them recipes for how to use Microsoft Word you run into the issue of which particular recipes they will need to know. What if they want to format their headers and footers like this? What if they want to auto-generate their table of contents, or format a table in a particular and slightly irregular way? Since each task is a recipe to be memorised by rote, each and every task has to be taught independently. The same sort of process occurs in mathematics education. It is particularly bad in the United States where each state and school district has their own ideas as to what are important facts that must be on the syllabus. The result is that textbook writers, in an effort to cater to everyone, touch on almost everything, but provide good coverage of almost nothing: the syllabus is a mile wide, but a mere inch deep. The same can be said of history, which also suffers a certain confusion of facts with the subject itself. The result is history textbooks that seek to cover vast stretches of history, but will often have little more than dates and a few scant details for any given event: history, and any in depth of analysis of any event and its impacts, is lost in the effort to try and touch on all the facts that have been deemed important. In mistaking a subject for a mere collection of facts, and in trying to cover all the bases, we have spread the syllabus so thin as to be bordering on useless.
The facts of mathematics are simply necessary preliminaries, just as names and dates are necessary preliminaries for the study of history. The subject of mathematics is itself much deeper and much richer than those simple facts. It is the art of abstraction, it is about structured reasoning, and reasoning about structure. To fail to understand and learn this depth is to fail to understand and learn the very subject itself.