Academic journal article The Review of Metaphysics

Thomists and Thomas Aquinas on the Foundation of Mathematics

Academic journal article The Review of Metaphysics

Thomists and Thomas Aquinas on the Foundation of Mathematics

Article excerpt

SOME MODERN THOMISTS, claiming to follow the lead of Thomas Aquinas, hold that the objects of the types of mathematics known in the thirteenth century, such as the arithmetic of whole numbers and Euclidean geometry, are real entities. In scholastic terms they are not beings of reason (entia rationis) but real beings (entia realia). In his once-popular scholastic manual, Elementa Philosophiae Aristotelico-Thomisticae, Joseph Gredt maintains that, according to Aristotle and Thomas Aquinas, the object of mathematics is real quantity, either discrete quantity in arithmetic or continuous quantity in geometry. The mathematician considers the essence of quantity in abstraction from its relation to real existence in bodily substance. "When quantity is considered in this way," he writes, "it is not a being of reason (ens rationis) but a real being (ens reale). Nevertheless it is so abstractly considered that it leaves out of account both real and conceptual existence." Recent mathematicians, Gredt continues, extend their speculation to fictitious quantity, which has conceptual but not real being; for example, the fourth dimension, which by its essence positively excludes a relation to real existence. According to Gredt this is a special, transcendental mathematics essentially distinct from "real mathematics," and belonging to it only by reduction.(1)

Jacques Maritain read the works of Gredt, including his Elementa, and in his magistral Degrees of Knowledge he agrees with Gredt that at least the objects of Euclidean geometry and the arithmetic of whole numbers are entia realia in distinction to the objects of modem types of mathematics. which he calls entia rationis. The objects of the former types of mathematics. Maritain says, are real in the philosophical sense that they can exist outside the mind in the physical world, whereas the objects of the newer types of mathematics cannot so exist. A point. a line. and a whole number are real beings, but not irrational numbers or the constructions of non-Euclidean geometries.(2)

Delving more deeply than Gredt into the nature of mathematics, Maritain stresses that when the mathematician conceives his objects they acquire an ideal purity in his mind which they lack in their real existence. By abstracting these entities from the sensible world the intellect idealizes them in such a way that not only their mode of being but their very definition is affected. There are no points, lines, or whole numbers in the real world with the conditions proper to mathematical abstraction.(3) Maritain also describes the purification or idealization of quantity in mathematics as a construction or reconstruction. He writes in his Preface to Metaphysics, "Quantity [in mathematics] is not now studied as a real accident of corporeal substance, but as the common material of entities reconstructed or constructed by the reason. Nevertheless even when thus idealized it remains something corporeal, continues to bear in itself witness of the matter whence it is derived."(4) He makes the same point in the Degrees of Knowledge: "In . . . mathematical knowledge, the mind grasps entities it has drawn from sensible data or which it has built on them. It grasps them through their constitutive elements, and constructs or reconstructs them on the same level. These things in the real [world] (when they are entia realia) are accidents or properties of bodies. but the mind treats them as though they were subsistent beings and as though the notion it makes of them were free of any experimental origin."(5)

In The Philosophy of Mathematics Edward Maziarz sees the essence of the scientific habit of mathematical abstraction as the mind's becoming "conformable and identifiable to the nature of substance solely as quantified."(6) He hastens to assure us that the objects of mathematics are not completely discovered in nature, as the scientist discovers the properties of the elements, but neither are they pure products of the mind. …

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