Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa

Introduction

The East-West Civilizational Clash in Mathematics: Pram?na versus Proof

In Samuel P. Huntington's terminology of a clash of civilizations, one might analyze the basis of the East-West civilizational clash as follows: the Platonic tradition is central to the West, even if we do not go to the extreme of Alfred North Whitehead's remark characterizing all Western philosophy as no more than a series of footnotes to Plato. But the same Platonic tradition is completely irrelevant to the East.

In the present context of mathematics, the key issue concerns Plato's dislike of the empirical, so the civilizational clash is captured by the following central question: can a mathematical proof have an empirical component?

The Platonic and Neoplatonic Rejection of the Empirical

According to university mathematics, as currently taught, the answer to the question above is no. Present-day university mathematics has been enormously influenced by (David Hilbert's analysis of) "Euclid's" Elements. Proclus,¹ a Neoplatonist and the first actual source of the Elements, argued that

Mathematics ... occupies the middle ground between the partless realities ... and divisible things. The unchangeable, stable and incontrovertible character of [mathematical] propositions shows that it [mathematics] is superior to the kinds of things that move about in matter.... Plato assigned different types of knowing to ... the ... grades of reality. To indivisible realities he assigned intellect, which discerns what is intelligible with simplicity and immediacy, and ... is superior to all other forms of knowledge. To divisible things, in the lowest level of nature, that is, to all objects of sense-perception, he assigned opinion, which lays hold of truth obscurely, whereas to intermediates, such as the forms studied by mathematics, which fall short of indivisible but are superior to divisible nature, he assigned understanding.

In Plato's simile of the cave, the Neoplatonists placed the mathematical world midway between the empirical world of shadows and the real world of the objects that cast the shadows. Mathematical forms, then, were like the images of these objects in water—superior to the empirical world of shadows but inferior to the ideal world of the intellect, which could perceive the objects themselves.

Proclus explains that the term "mathematics" means, by derivation, the science of learning, and that learning () is but a recollection of the knowledge that the soul has from its previous births, which it has forgotten—as Socrates had demonstrated with the slave boy. Hence, for Proclus, the object of mathematics is "to bring to light concepts that belong essentially to us" by taking away "the forgetfulness and ignorance that we have from birth" and reawakening the knowledge inherent in the soul. Hence Proclus values mathematics (especially geometry) as a spiritual exercise, like hatha yoga, which turns one's attention inward, away from sense perceptions and empirical concerns, and "moves our souls toward Nous" (the source of the light that illuminates the objects of which one normally sees only shadows and which one could better understand through their reflections in water).

In regarding mathematics as a spiritual exercise that helped the student to turn away from uncertain empirical concerns to eternal truths, Proclus was only following Plato. The young men of Plato's Republic (526 et seq.) were required to study geometry because Plato thought that the study of geometry uplifts the soul. Plato thought that geometry being a knowledge of what eternally exists, the study of geometry compels the soul to contemplate real existence; it tends to draw the soul toward truth. Plato emphatically added, "if it [geometry] only forces the changeful and perishing upon our notice, it does not concern us,"² leaving no ambiguity about the purpose of mathematics education in the Republic.

Rejection of the Empirical in Contemporary Mathematics

A more contemporary reason to reject any role for the empirical in mathematics is that the empirical world has been regarded as contingent in Western thought. Any proposition concerning the empirical has therefore been regarded as a proposition that can at best be contingently true. Hence, such propositions have been excluded from mathematics, which, it has been believed, deals...