Making Introductory Mathematics Worthy of the Name Mathematics

Dr. Jonathan Kenigson, FRSA

Senior Fellow of Natural Philosophy

Athanasian Hall, Cambridge LTD

130 Old Street

London, UK

EC1V 9BD

The clamor of freshman feet at every university in the land heralds new excitement and untold journeys of wonder through the corpus of Greats. Students may enroll in Literature, in which they imbibe the sorrows of Shakespeare’s Caesar, the perilous cunning of Milton’s Satan, the mockery of Aristophanes, and the wisdom of Plato. They may enroll in Art, in which they learn of the saturnine genius of masters and the techniques of color, contrast, and perspective. In History – if it is taught with proper sources – they may find some appreciation of the policies and precepts that have fashioned this republic, as well as the nature of the Constitution and its guarantees. They may learn of wars and rumors of wars; of heroes known and unknown who have been resolved to dust by the train of ages, only to be resurrected to modernity by the diligence of some scholar in a specialist pursuit. The greatest historian is a poet of the requiem. For Tacitus, Plutarch, Livy, Thucydides, Gibbon, the end is as fraught with the beginning as the scourge of pride is to the inevitability of falling. Republics and empires are built on the crags of thousand-story mountains, stolen away not by Vandals, Goths, and decadent scandal, but by those who – ages later – refuse to reckon or remember what was, what could have been, or what ought to be. Other subjects die not by refusal to remember but refusal to concede that remembrance has any value whatsoever. Mathematics is one of these subjects. It is the stem of STEM, and yet is masqueraded as an entity altogether different from its customary forms. As with most common crimes of theft of identity, it is prudent to account the recollections of the victims.

It is perhaps predictable, if not lamentable, that so-called university mathematics is rarely the most desirable choice of the college freshman. Such person arrives hopelessly immured in the unspoken notion that mathematics – unlike History or Art – is not a struggle of empires or a tutelage of masters, but rather a valiant effort against inanity. Mathematics is supposed to be immutable and unchanging. By this fact, the presentation of mathematical concepts is falsely taken to be similarly immutable and unchanging. In today’s university, so-called comprehension of mathematics is to be won through the regurgitation of formulae taught anonymously by some pre-packaged learning management system. Such systems feature automatically-graded quizzes, copious homework exercises, and high-stakes tests that test, in the best case, what the homework “taught” and the quizzes “quizzed”.  Applied questions test the arrival times of boats in the water; trains passing each other with some delay; the number of animals in a herd given the number of heads and the number of feet; and other topics of obviously eminent importance. In the best case, a student passes the course only to pass on the bitter hatred for so-called mathematics that pervades this country. Should one be surprised that this student is bitter, or stops attending altogether? Having quaffed the sweeter nectars of History, Politics, Music, Philosophy, and Art, should one be surprised that this person should not willfully settle without complaint for the half, or even the quarter? 

Many students believe that university mathematics as it is so taught is useless. They are only half correct in this assessment, as such a subject – should it be given the dignity of such designation –  is both useless and dangerous. There is no greater driver of innumeracy than the graduate who despises a crucial topic so thoroughly that they choose to resolutely forget it, or even to remember its pedantries incorrectly. Why should students be deprived of the Quadrivium, which informed mathematical knowledge for ages, and upon which all modern Analysis, Algebra, Geometry, and Mechanics is based? The Quadrivium encompasses Arithmetic, Astronomy, Geometry, and Music. These are the four disciplines that – in addition to Grammar, Logic, and Rhetoric – comprise the classical Liberal Arts. Thus, when one says that they should major in Liberal Arts, one should declare a minor in pure mathematics at minimum. Students should be taught the fundamentals of these arts from the texts that informed their historical evolution and the problems upon which the texts sought to expound.

Let us consider Arithmetic. In the first instance, the study of number as discrete quantity is discussed by many great authors who wrote on other topics of crucial importance for civilization. The nature of proportion, for instance, is found in the Phaedo and Theaetetus, which also explore important themes of the soul. Such discussion is continued through Aristotle’s Categories and Book 5 of the Metaphysics, as well as Nicomachean Ethics and the Politics, where proportional theories of virtue and virtue-ethics are considered.  More overtly mathematical discussions of proportion are found in the Archimedean corpus (especially Sphere and Cylinder and Spirals); in Book 2 of Nicomachus’ Arithmetic; Book 3 of Descartes’ Geometry; and Newton’s Principia. The distinction between number and other manners of quantity are explored in Kant’s Critique of Pure Reason and also the Critique of Judgment. The latter is an exposition of aesthetics – a discipline traditionally enriched and informed by the study of proportion. From this juncture, discrete number appears as an aspect of Relativistic Mechanics as well as the modern theory of number crafted by Hardy and Ramanujan, among others. Discrete number also appears in the theory of infinite series, upon which all modern computation and the entire theory of approximation of functions rests. Why discuss such notions in any depth when one can work the same exercise in thirty different ways in front of a computer? Is not the latter more informative, more edifying? Does it not satisfy some innate urge for discovery not quenched by the minds of masters who dared to be original and robust in their thoughts? What stories dead books can tell, if living professors choose to resurrect them! Let us not forget books of modern mathematics written by practitioners of the subject, which are far more subtle and technical.

No student is too ignorant to be rendered moreso by the emaciation of a beautiful and subtle subject for the express purpose of expediency and uniformity of exposition. The goal of the Liberal Arts is to liberate, not to enslave or render the student bitter. Mathematics should be taught – as other subjects should of right be taught – as a discourse of minds in which students are participants. If this mathematics is useless, it is at least useless and edifying. Whatever politician of whatever persuasion chooses to read this piece would do well to consider that great texts of politics are often great texts of mathematics. Theories of justice, equity, and elections are governed by the same laws that predict the orbits of moons and the operation of computers. It is better that students learn these laws than the gaunt forms to which they have become accustomed to despising so thoroughly.

Dr. Jonathan Kenigson, FRSA

SDG