The Place of Mathematics in a Liberal Arts Education
Although it is perhaps not altogether clear what the terms ‘liberal arts education’ or ‘humanistic education’ mean, the usual, perhaps hazy, accounts of these phrases have tended to reinforce the divide between, for example, the ‘arts’ area of the curriculum and the technical area, i.e. science and mathematics. Thus it is not unusual to find in our classes very gifted students for whom the very notion of mathematics is forbidding. Some of these students have been successful at every other subject they have undertaken except mathematics, and express their intense dislike for the subject in no uncertain terms.
The situation alluded to above has always been a source of disappointment to me, since before my studies in Philosophy, I taught mathematics at the University level. (It may perhaps still be true to say that mathematics is my favorite subject.) So for me a key question arose: How could it be that so many of the able students I had the opportunity to teach claimed to hate a subject which I held in such high regard? I got a chance to try and answer this question when it became necessary for all the Liberal Arts programs at the College level in Quebec to start teaching a course called The Principles of Mathematics and Logic. Given my background, I chaired the committee which generated the course description for this new part of the curriculum. While writing it, I thought that it would be able to serve as a test of the following idea: Since mathematics is such a wonderful subject, intelligent people who think they hate it may not understand what the true nature of mathematics is. If this could be made known to them, their attitude might change. What follows is an account of what I take the ‘true’ nature of mathematics to be, and a justification of why it is that an understanding of mathematical discourse should be an essential part of any Liberal Arts curriculum, as well as an introduction to some of the main ideas featured in the text.
The core concept around which the teaching of mathematics in a Liberal Arts curriculum should be centered is the idea of a proof. The view of mathematics as a discipline whose essential function is to produce proofs is eloquently put forward by the British mathematician G.H. Hardy in his book, A Mathematician’s Apology. In this book, Hardy says that, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because it is made with ideas.” Hardy emphasizes the beauty of what he calls pure mathematics as opposed, for example, to applied mathematics as it is employed in the natural sciences, as well as mathematics of the puzzle-solving variety, e.g. problems in chess. And he is at pains to claim that pure mathematics is not ‘useful.’ I agree with Hardy that to understand what mathematics is, one must understand the particular nature of the patterns that mathematicians produce. However, I disagree that the understanding of these patterns is not ‘useful’, and that they should be studied only by mathematicians. The rest of this preface argues that once we understand the nature of these patterns, it becomes evident why mathematics should have a place in a Liberal Arts education. These patterns are called proofs. To make this concept clear, the first point to note is that a proof is, or plays a role in, a kind of discourse. (It is this fact which links mathematics with what might be called the ‘critical thinking’ view of Liberal Arts curricula.) In this regard it is appropriate to note that the record of this discourse is called a literature. To have an understanding of, or even to participate in this literature, what is required?
First we must realize that the structure of a proof exhibits certain distinctive reasoning patterns. They are (kinds of) arguments. So to understand these patterns we must do some logic, the subject one of whose main burdens is to describe and evaluate arguments. This is done by taking up the following questions: What is reasoning? What is logic? What is deductive logic? What is inductive logic? Do both deductive and inductive arguments play a role in mathematics? What makes valid arguments valid? (The formal character of deductive validity.)
After claiming that a proof is a kind of valid deductive argument, we present the view that proofs are presented via a structure called an axiom system. This idea involves talking about whether or not we should require that those statements from which we reason, and for which we do not require support, called axioms or postulates, should be self-evident or be in some other substantial sense obviously true. We also take up the idea of a rule of inference, which is mentioned in the part of the course, described above, having to do with logic. After the idea of an axiom system is developed adequately, we give examples of such systems; e.g. an axiomatic development of the ordinary number system and parts of elementary algebra; an axiomatic development of Euclidean and /or non-Euclidean geometry; an axiomatic development of probability theory, including set theory, along with statistical (inductive) applications. The students are asked not only to recognize proofs in these examples, but to actually construct proofs on their own. It may seem to some of you that this is an unreasonable demand to make on our students, especially the ones who claim to hate math. But I have found just the opposite. Once it is made apparent that mathematics essentially deals with the constructing of proofs and not with, as many of our students think, memorizing unjustified calculative schemes, and that constructing proofs requires imagination and ingenuity, and that successful proofs are powerful in that they can demonstrate that a whole class of problems can be looked at and solved in one relatively short demonstration, many of them change their attitudes toward the subject.
In answer to the question, ‘Why should a mathematics course such as the one described above be included in a Liberal Arts curriculum?’, the following are points are relevant:
1. As alluded to above, one idea of what a Liberal Arts program should be is essentially tied to the view that perhaps the main goal of such a program is to produce ‘critical’ thinkers. On this view, an exposure to mathematics is obviously justified, since critical thinking essentially involves sensitivity to argument, and mathematics provides the context for presenting and scrutinizing arguments according to the most exacting standards of logical correctness. In coming to understand this, the students also gain an appreciation the power of abstract thought and the elegance which it often embodies.
2. Another idea of what a Liberal Arts program should be is sometimes expressed as the ‘great books’ tradition. On this view, the idea is to expose the students to the lasting contributions of , e.g. Western literature. But here also, surely, mathematics has a place. It is hard to argue that understanding a proof of the Pythagorean theorem, or Euclid’s proof that there are infinitely many prime numbers, or Wiles’ proof of ‘Fermat’s last theorem’ are any less valuable a contribution to the ‘canon’ than any of the other contributions usually mentioned. Since a course as described above involves doing mathematics, i.e. constructing proofs rather than just criticizing them, some of the students leave the course with the feeling of a mastery over a (small) area of mathematics. This is, I think, somewhat akin to what it feels like to have acquired virtuosity in music. Although the students are not constructing deeply original proofs, they come to ‘own’ the proofs they understand in the same way a singer ‘owns’ a song they have mastered, even if it was not written by them.
3. Finally, it is liberating to understand what those on the ‘other side’ have been up to all these years. It creates the possibility for constructive conversation between the ‘arts’ and ‘science’ types (mathematics is, after all, the language of science) which can only be productive for everyone concerned.