Oxford Figures, Chapter 1: 800 years of mathematical traditions

The logical tradition

The history of logic has long been intertwined with that of mathematics, while never quite forming the same area of inquiry. In the great medieval pedagogical classification of subjects into the trivium and the quadrivium, logic, or `dialectic', was part of the trivium, alongside grammar and rhetoric, to be studied before and more prominently than the four mathematical subjects of the quadrivium--arithmetic, geometry, music, and astronomy. At Oxford, logic has been taught throughout the history of the University, sometimes by the same people as taught mathematical sciences. At the research level, the great Mertonians of the fourteenth century, such as Thomas Bradwardine, Richard Swyneshead, William of Heytesbury, John Dumbleton, and Richard Billingham, reached unparalleled heights of sophistication and insight into logical matters, arguably not to be reached again for another five centuries, and by the end of the century their influence had spread across Europe.

In the succeeding centuries logic continued to be taught as a fundamental part of the general education which Oxford hoped to instil in all undergraduates. In the words of John Wallis, the Savilian Professor of Geometry who himself wrote a textbook on logic in 1687, the purpose of teaching logic was to lay `the foundations of that learning, which they are to exercise and improve all their life after', explaining its merits as being

to manage our reason to the best advantage, with strength of argument and in good order, and to apprehend distinctly the strength or weakness of another's discourse, and discover the fallacies or disorder whereby some other may endeavour to impose upon us, by plausible but empty words, instead of cogent arguments and strength of reason.

For almost as long as logic has been taught at Oxford, there have been complaints about it from students, both during their studies and in retrospect. Besides perennial charges that it was too dry and boring, a long-debated point of substance was whether logic and mathematics were both necessary for mind-training purposes, or which was preferable. An anonymous writer in 1701 (possibly the 1690s undergraduate John Arbuthnot) clearly differentiated what he saw as the advantages of each, with the moral victory to mathematics:

Logical Precepts are more useful, nay, they are absolutely necessary, for a Rule of formal Arguing in public Disputations, and confounding an obstinate and perverse Adversary, and exposing him to the Audience or Readers. But, in the Search of Truth, an Imitation of the Method of the Geometers will carry a Man farther than all the Dialectical Rules.

While the pedagogical arguments between logic and mathematics continued for another two centuries, mathematics gradually superseded logic, in effect, as an instrument for training the mind (and strong views were expressed, too, on behalf of classics as the only effective mind-sharpening study). By the late nineteenth century, Oxford students who were in line for a pass degree had, besides offering Latin and Greek, to choose between mathematics (quadratic equations and some propositions from Euclid) and logic. A guide for undergraduates advised caution in making the choice:

Logic will, most probably, be untried ground to the student, and we advise him to consider well before he chooses this in preference to Mathematics ... he must be prepared for hard, and, perhaps, uninteresting reading, demanding not only close attention, but fair powers of memory.

The late nineteenth century also saw a resurgence of interest in logical research, to which the Christ Church mathematics lecturer, Charles Dodgson (Lewis Carroll), made notable and influential contributions, sometimes underrated on account of their apparent whimsicality. In exploring new logical ideas as well as how to teach logic more effectively, Dodgson's work can be seen to lie within a long tradition of Oxford logical activity. Throughout the twentieth century logic continued to form a part of Oxford concerns, moving more into the area of mathematical philosophy.

next section, previous section