Matter Matters: Metaphysics and Methodology in the Early Modern Period (review)

This is a puzzling book, alternately bewildering and informative. Smith's aim is to investigate seventeenth-century answers to the question of why there should be a material world. The answer, he maintains, shows a deep connection between mathematics and matter. In Smith's telling of the tale, seventeenth-century thinkers (most notably Descartes and Leibniz) were led to affirm a striking biconditional: mathematics is intelligible if, and only if, matter exists. If this central thesis is correct, much scholarship on seventeenth-century philosophy must be revised.

The received view is that Descartes and Leibniz accepted the commonplace that mathematical truths are necessary and independent of the structure and contents of the material world. On this quite traditional way of thinking, a Euclidean theorem is both true and intelligible, regardless of what kind of world God might have created, including one without material bodies or with a non-Euclidean spatial structure. Descartes surely seems to accept the traditional view when he writes that truths concerning the "immutable nature" of the triangle are demonstrable, whether or not such a figure exists "outside my thought" (AT VII:64). Likewise, Leibniz is typically read as holding that the truths of mathematics hold in all possible worlds (including, presumably, worlds without matter), so that mathematical truths are independent of material considerations.

Perhaps the received view is in error, in which case scholars would be very much in Smith's debt. I think, however, there is no compelling evidence to believe that either Descartes or Leibniz held that mathematics requires matter. A brief examination of the evidence in the case of Descartes will show the severe limitations in Smith's central thesis. I leave aside an examination of the case of Leibniz, although his complex and changing view of the "ideal" nature of mathematics (distinct from both the substantial and phenomenal levels) are prima facie committed to the independence of mathematics from matter.

So far as I can determine, Smith has found only two passages that might directly commit Descartes to the claim that mathematical intelligibility requires matter. The first is from Principles II.8, where Descartes says, "A continuous quantity of ten feet cannot be understood without some extended substance of which it is the quantity, although it can be understood without this determinate substance" (AT VII:44-45). Smith takes this to commit Descartes to the thesis that number in general cannot be understood without extended substance, so that all of mathematics depends upon the existence of res extensa or matter. But this reading misconstrues the passage and its context. Principles II.8 bears the title "Quantity and number differ only in thought from that which has quantity and is numbered" (AT VII:44). The relevant distinction is between quantity on the one hand, and number on the other. Descartes' terminology differs somewhat from the tradition, but he takes geometry to investigate continuous quantities (what the tradition termed "magnitudes"), while arithmetic is concerned with discrete numbers ("multitudes" in the tradition). His point is that we cannot conceive of an extended quantity (line, angle, surface, or solid) without supposing some underlying extended stuff. Likewise, we cannot conceive of a number "in itself" apart from some numbered multitude. However immaterial, extensionless things like angels or human souls can surely be numbered. Thus, Descartes does not require matter in order for mathematics to be intelligible.

Smith also interprets Descartes' remark in a 1638 letter to Mersenne as claiming that mathematics requires matter. Reporting an objection of Roberval, Mersenne wrote that "having supposed that God created nothing, [Roberval] claims that there would still be the same three-dimensional [solide] real space there is now, and he bases the eternal truth of geometry on this space" (AT II:117). Descartes responded that "not only would there not be any space, but even those truths we call eternal (such as that the whole is greater than the part, etc.) would not be truths, if God had not...