"The Study of Mathematics," pages 58-73
Deep contemplations lead to beautiful edifices, but that beauty is remote to the beginner and its achievement and appreciation require hard-to-obtain knowledge. “Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple [pages 58-59].” Their students see only the steep upwards path, and not the gorgeous structure at the end.
In terms of concealing the ultimate purpose, mathematics education might be in even worse shape than classics. The significance of mathematics is often couched in how it leads to better machines, improved transport, and military might. Of course, the limited mathematics training that most people receive does not conduce to these ends. Why do they study mathematics? The typical response is that the study of math enhances the ability to reason – though this response is made primarily by people who themselves teach all sorts of fallacious nonsense. Improved reasoning itself is viewed as contributing to prudent personal decision-making: hardly a goal worthy of teaching mathematics to all educated people. But Plato understood that mathematics is requisite for the apotheosis of mankind.
Mathematics pairs truth with unvarnished beauty; it offers respite from the painful compromises of our quotidian existence. The beauty of mathematics is the result of rigorous logic, not a product of any conscious aesthetic design.
How can this beauty, this higher quality of mathematics, be communicated by teachers? In geometry, it is best to focus, at first, not on theorems and proofs, but on diverse illustrations (of triangles and their associated lines, meeting in a single point, for instance). “In this way belief is generated; it is seen that reasoning may lead to startling conclusions, which nevertheless the facts will verify; and thus the instinctive distrust of whatever is abstract or rational is gradually overcome [p. 62].” The concrete is the way to the general.
For new learners, algebra is hard to comprehend, with its reliance on letters (variables) instead of numbers. But algebra presents truths that go beyond the particular, general truths, and it is these that allow “the mastery of the intellect over the whole world of things [p. 63].” Teachers tend to fail to impart an understanding of the principles brought to bear in algebra, however, even if students learn recipes to apply rules that produce correct answers.
After algebra, it is dealing with infinity (as in the infinitesimal calculus) that presents the next hurdle. “The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our own age has to boast [p. 64].” Cantor and Dedekind showed that when dealing with infinity, you could remove some elements from a set, and still have the same number of elements in the set. This notion cleared up all of the misconceptions concerning infinities, and has opened up grand new vistas of thought. In the past, much of the foundation of mathematics was clearly fallacious, a practical compromise that mixed logic with superstition; now, the need to compromise has been expunged. Pure mathematics, mathematics as an end in itself, can be built from first principles that themselves can survive intense scrutiny.
Textbooks typically fail to convey the unity and purposeful progression of mathematics. But the beauty and drama of mathematics lie in its interconnections, in the relation of many propositions to a few fundamental ideas. Learners must not be distracted from these core notions through a plethora of inessential or unconnected examples.
The ultimate unifying discipline within mathematics is symbolic logic, which is a product largely of the nineteenth century and still developing today. “The true method of discovery in symbolic logic, and probably also the best method for introducing the study to a learner acquainted with other parts of mathematics, is the analysis of actual examples of deductive reasoning, with a view to the discovery of the principles employed [p. 67].” The realization that only a few simple concepts underlie mathematics, and that those concepts are regularly (but perhaps unconsciously) employed in thinking, can be transformative. What had been glimpsed is now seen clearly, and the panorama is beautiful.
Symbolic logic has replaced the former foundations of mathematics, which were philosophical, and thus uncertain. The sounder base has rendered the superstructure more intellectually pleasing, and this pleasure should be made available to students.
Logic and mathematics exist outside of humans and their thoughts – but we can still appreciate the beauty of mathematical objects, whether they be our creations or our discoveries.
For students, the goal shouldn’t be just to inform them of conclusions, of the end points of a chain of reasoning, but also to take them along the most splendid path to those ends. “An argument which serves only to prove a conclusion is like a story subordinated to some moral which it is meant to teach: for aesthetic perfection no part of the whole should be merely a means [p. 70].” Elegance and generality in a mathematical argument, a proof, derive from using only the most fundamental, minimal assumptions necessary to reach the conclusion.
The common notion that truths are relative, that one person’s truth need not be another’s, and that there is no impartial standard to decide the matter, meets its demise in the arena of mathematics.
Mathematics needn’t rely on its practical effects for its justification. But in a world full of injustice, sometimes it seems hard to countenance a life spent in thought, aloof from the evils of the world, while enjoying a beauty that is not available to most people. Of course we need some people to “keep alive the sacred fire [p. 72],” but this rationale might seem inadequate in the face of current troubles. Here, the practical applications of mathematics can help, by reminding us that cloistered study in the near term can lead to tremendous improvements in human happiness down the road. Could we harness steam power or electricity without the development of mathematics? Of course, we cannot know what sorts of mathematics will lead to the best innovations in the future, so we should avoid investing solely in those branches of mathematics that have proven useful in the past.
The love of truth holds the power to raise our moral existence, “and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith [p. 73].” Cloistered study is its own reward, but it also ennobles our minds; the teaching of mathematics should keep in mind this indirect benefit.