# Introduction

Isaac Newton’s *Mathematical Principles of Natural Philosophy* embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of the mathematical sciences.

Barrow’s attitude towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. ^{1} In his last years he became a prolific **[End Page 265]** author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. ^{2} It is on certain of his sermons, as well as on the philosophical discussions contained in the *Mathematical Lectures* that our account of Barrow’s philosophy of the mathematical sciences will rest. ^{3} Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.

It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s *Principia Mathematica.* In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.

During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian **[End Page 266]** mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s *Mathematical Lectures* (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.

# Barrow on Matter and God

Recent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His...