Philosophy of mathematics, which might seem like some boutique academic specialty, has played a remarkable role in the history of Western thought. To Plato, for example, mathematics provided the very model of knowledge, of truth apprehended with certainty not by the senses but by the mind. St. Augustine learned that lesson from the neo-Platonists of his day, which allowed him to take a crucial step toward his religious conversion, for it made intelligible the possibility of a non-material god. “How is pure mathematics possible?”—the famous question at the heart of the Critique of Pure Reason—was Kant’s way of asking in an aphorism what the world must be like if it can be described by mathematical physics. Mathematics raises, in an acute way, the question of how (or whether) we can bridge the gap between our knowledge and the objects of our knowledge.
The mathematician and philosopher James Franklin is a leader of the “Sydney School,” which has developed an account of mathematics that he sets out in An Aristotelian Realist Philosophy of Mathematics. “Aristotelian” means not that it strictly follows or develops Aristotle, but that it is recognizably in the same ballpark. He presents it as a middle way between two poles—broadly classifiable as Platonist and nominalist—that have dominated the subject and dictated the terms in which it is discussed. He therefore begins with an ancient question about the status of universals, those general properties (such as shape, number, mass) whose relationships are the proper subject matter of systematic knowledge.
For a hardcore Platonist, universals—which include the objects of mathematics—constitute a realm of entities that exist independent of minds and outside space and time. Platonist views of mathematics have had an enduring appeal. The mathematician hunkered in a foxhole, earning his pay, finds it difficult to set aside the prejudice that he is grappling with something real—to keep up morale, if nothing else. And if mathematics concerns unchanging entities detached from the messy and confusing world of the senses, we can begin to explain and justify the belief that mathematical knowledge is necessary and certain.
To a nominalist, only particular individuals exist. Words purporting to denote universals are conventional labels. Nothing real corresponds to the boundaries they draw, which are nothing but convenient ways of organizing our experience (thus far). Franklin holds that universals are real, not because they inhabit some special realm of their own but because they refer to real properties of things. What universals exist can be discovered by investigation. Blue is one, goes the argument, because there is an ensemble of physical properties one of whose effects is that bodies which possess them will, in appropriate circumstances, look blue. The facts that the normally sighted make similar judgments about what is blue and can agree about how to line up shades of blue along the color spectrum testify that blue corresponds to something real. The inability of different people to compare their subjective experiences of color is beside the point.
Mathematics presents difficulties for each of these views. A Platonist, for example, must explain how we physical creatures can have access to a world of non-spatial, nontemporal entities and how knowledge about that other realm could be relevant to the world we live in. Applied mathematics is also a challenge to nominalists: Why are propositions about things that do not exist so uncannily useful? Aristotelian realism, which places applied mathematics at the center of its account, must explain universals that are “uninstantiated.” Many notions of mathematics, such as high-order infinities, do not seem to be realized in the physical world; indeed, if the world is finite, even large numbers are not the numbers of anything. In what sense, then, can these be “real properties of things”?
Part I of Franklin’s book, “The Science of Quantity and Structure,” argues that Aristotelian realism is adequate to give an account of mathematics and that this account, uniquely, justifies two striking claims: that the objects of mathematics include real properties of the physical world and, accordingly, that “there are necessary mathematical truths literally true of physical reality.” Part II, “Knowing Mathematical Reality,” ranges widely over questions of epistemology, such as understanding, visualization, explanation, and non-deductive reasoning (evidence other than proofs for or against mathematical assertions). It includes an outline for a program to explain how, building on inherent mathematical abilities—such as babies’ perceptions of multiplicity—we can acquire higher-level knowledge of abstract and unperceived mathematical structures.
“Quantity” begins with the counting numbers: one, two, three . . . Any account of them must begin with Gottlob Frege’s tour de force, The Foundations of Arithmetic (1884). It is a deep inquiry into the notion of number, a brilliant demolition of competing theories, a founding document of analytical philosophy, and a treat for anyone capable of enjoying a first-rate mind at work.
A number, Frege says, cannot be a property. What would be the number of the Iliad? One (poem)? 24 (books)? 15,693 (verses)? To count we need not just the stuff being counted but also a “criterion of identity” that specifies what it is about that stuff that one should count—for example, the criterion of “being a poem” or of “being a book of a poem.” Frege concludes that since a number is not a property—nor, he also argues, a mental state—it must be a (nonspatial, nontemporal) thing.
Not so fast, says Franklin. If we accept the reality of universals, including those that express relations, there is a third possibility, one that doesn’t require belief in mysterious Platonist objects: The number of verses in the Iliad is the relation between the Iliad and the universal “being a verse.” As Frege says, however, we must give an account not only of numbers (“There are three cows in the barn”), but also of number theory (“For every n there is a prime number greater than n”). Franklin does not address that explicitly. I suspect he would respond that number theory is about the structure of arithmetic; in his account, structure requires no Platonist objects either.
Much advanced mathematics concerns non-quantitative notions—such as symmetry and continuity—that have been called structural. We may know it when we see it, but structure is difficult to define in the abstract. Franklin offers the following: Structure is what can be defined in terms solely of part/whole relations and logic. In particular, structure is about how things are related, regardless of what those things are. (A highly technical aside: Second-order logic is required to characterize many important mathematical structures; that potentially requires some hefty philosophical and mathematical commitments that one might prefer to avoid when laying a subject’s groundwork.)
Everything rests on the contention that mathematical entities can be literally exemplified in the world. Franklin’s running example is the structure of a graph. Leonhard Euler, the greatest mathematician of the eighteenth century, introduced the notion of a graph to solve the Königsberg Bridge Problem. Through the city of Königsberg runs the river Pregel, which contains two islands. The city had seven bridges, each connecting an island either to one bank of the river or to the other island. It was believed impossible to take a walk that crossed every one of the bridges exactly once; and Euler proved this was true by proving a fact about a certain graph. Franklin argues that the city of Königsberg literally has the structure of that graph and that Euler’s proof therefore demonstrated a necessary truth about the physical world.
A graph may be visualized as a collection of dots on paper (call them nodes) and line segments joining some of the pairs of points (call these lines edges, and call the points an edge joins its end points). The structure of a graph is a universal—the end-point relationship among nodes and edges. If we say that each of Königsberg’s four land masses (two river banks, two islands) counts as a node, that each bridge counts as an edge, and that the end points of a bridge are the land masses it connects, we have identified a graph structure of which, says Franklin, Königsberg is literally an instance. Euler proves that there is no path in that graph—no way to proceed successively from one node to another by following edges that connect them—on which every edge appears exactly once.
An obvious objection is that the example seems to have been cherry-picked. Königsberg might literally be a graph, but we often represent a complex situation with some structure that has been deliberately and radically simplified—for example, treating a fluid made of particles as if it were a continuous substance. In that case the real world is not an instance of what’s analyzed and Franklin’s account seems to get no purchase. I’ll attempt an abbreviated version of his response. Consider a coin, whose exact outline is complex; it’s certainly not a Euclidean circle. The exact outline of the coin nonetheless realizes some mathematical structure and without leaving the world of mathematics we can establish necessary relations between it and other structures. For example, we can consider those structures that represent “a figure no more than x% out of round” and establish, by proof, upper and lower bounds for their areas. And it’s quite plausible to assert that some real coin literally exhibits the property of being “no more than five percent out of round.”
The ability to reason in this way about not-fully-known structures typically depends on the structures’ being “stable”—that is, having properties that are relatively insensitive to variations in the structure. Stability is a mathematical property that can be established by mathematical means.
It is striking that the landscape described more than fifty years ago in Stephan Körner’s well-known book Philosophy of Mathematics, An Introductory Essay looks so much like the one visible in the up-to-date Stanford Encyclopedia of Philosophy. The main non-Platonist contenders are listed, then and now, as: logicist (mathematics is nothing but logic), formalist (mathematics is the manipulation of symbols according to specified rules), and intuitionist (mathematics consists of constructions performed in the minds of mathematicians). Each has been philosophically fruitful and has motivated significant developments in mathematics. (I wonder if there is a field in which philosophy has had a more beneficial effect on practice.) And they remain at a stand-off, each able to present formidable criticisms of the others.
Franklin claims to offer a way out of the impasse and his book seems to me—a professional mathematician with an amateur interest in philosophy—an important one. A short review cannot do justice to the variety of problems considered and the interesting angles of attack offered by his realist point of view.
An epilogue, titled “Mathematics, Last Bastion of Reason,” places the discussion in a larger context. Readers of this journal needn’t be reminded of the fashionable anti-rationalisms and irrationalisms that have flourished in the arts and in the humanities. Franklin notes that popular versions of even hard sciences “have caught some unpleasant philosophical diseases”—as when quantum mechanics, which makes predictions verified to spectacular degrees of accuracy, is “coated in prose about ‘reality dependent on the observer.’ ” Mathematics, he says, has always provided support for views that “exalt the capacity of the human mind to find out the truth” and has been “a perennial thorn in the side of opinions that abase human knowledge, and claim it is limited by sense experience, cultural experience, or one’s personal education and perspective.” Franklin gets into the ring with Mr. Frege, et al. not only to hash out ancient philosophical disputes but to take a stand for the very possibility of objective truth.