‘Tis universally allow’d, that the capacity of the mind is limited, and can never attain a full and adequate conception of infinity: And tho’ it were not allow’d, ‘twou’d be sufficiently evident from the plainest observation and experience. (David Hume)
The eternal silence of these infinite spaces fills me with dread. (Blaise Pascal)
The mainstay of seventeenth-century continental philosophy was rationalism. Two of the three philosophers most frequently classified as rationalists, Descartes and Leibniz, have already made a large impact on this enquiry through their work in mathematics; the third, Spinoza, so revered mathematical method that he modelled his major work (his philosophical masterpiece Ethics) on Euclid’s axiomatization of geometry. This is revealing, because British philosophy, that same century and the next, saw a backlash, in the form of empiricism, and the central point of controversy between rationalism and empiricism was the extent to which understanding of the world could be arrived at by a priori means—by that exercise of pure reason which is characteristic of mathematics. The rationalists held that deep substantial truths about the structure of the world—not just mathematical truths—could be discovered in this way. The empiricists insisted that it was only through experience that we could come to know such truths. It is worth noting the similarities between the empiricists’ reaction to rationalism and Aristotle’s reaction to the more metaphysical strains in his predecessors. In this chapter I shall try to spell out the implications of this for continuing thought about the infinite.
In Chapter 3 we saw a growing friendliness towards the actual mathematical and the metaphysical infinite, which had begun to break the mould