Academic journal article The Review of Metaphysics

Franklin, James. an Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure

Academic journal article The Review of Metaphysics

Franklin, James. an Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure

Article excerpt

FRANKLIN, James. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure. New York: Palgrave Macmillan, 2014. x + 308 pp.--In the opening pages of his book, James Franklin declares that his intent is to show that mathematics is a science of the real world just as much as biology and the social sciences. Where biology studies living things and sociology studies human social relationships, mathematics studies the quantitative and structural-patterned aspect of things. For example, "a typical mathematical truth is that there are six different pairs in four objects. The objects may be physical, mental or abstract." The truth about pairs of objects is not hypothetical or logical, but a straightforward truth about objects of any kind; for relations are as real as colors or causes. Elementary mathematics, he explains, is the science of quantity, and higher mathematics is the science of structure. Chapters are devoted to both kinds of mathematics.

In defending his Aristotelian approach, Franklin targets both Platonism and nominalism. He criticizes Platonism for attributing to the objects of mathematics a reality necessarily abstract and separate from physical objects, and nominalism for regarding mathematics as having no real subject but being only a manner of speaking or making inferences concerning ordinary physical things. Nominalists hold that universals are not genuine constituents of reality--that the only realities are particular things. Franklin, to the contrary, insists that "Aristotelianism regards mathematics as literally being about some aspect of reality, about certain kinds of properties and relations rather than about individual objects." A principal strength of Aristotelian realism, he maintains, is that it can give a natural account of the difference between a law of nature and a mere coincidence. In defending his view, Franklin is necessarily led to a discussion of Aristotle's theory of abstraction and other epistemological issues. A physical law stating that copper conducts electricity or that water expands when frozen is the result of some real connection. The properties of copper and water and properties considered mathematical, like shape, size, symmetry, and continuity are categorical. They are not properly defined by how they would react in some counterfactual circumstance.

Philosophers of mathematics generally have been empiricist, in the style of John Stuart Mill and Imre Lakatos; either they deny the necessary certainty of mathematics, or admit its necessity but deny mathematics a direct application to the real world. An Aristotelian philosophy of mathematics, by contrast, finds necessity in truths that are directly about the real world.

In his review of higher mathematics in the nineteenth and twentieth centuries, Franklin finds that pure mathematics has been pushed into an ever more abstract and structural direction. …

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