Abstract

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 but real beings. 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 but a real being. 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.