## Mathematics and linguistics (part 2)

In part 1, I discussed the non-specialist’s experience with both mathematics and linguistics, and suggested that their experience is, in both cases, essentially prescriptivist in nature. Before discussing the relationship between these fields, we must move beyond the non-specialist’s perceptions and understand more about the actual scope of each of these fields. In this part, I’ll address the question of mathematics.

The definition of mathematics offered by the OED (which I discussed here) proposes that modern mathematics is “the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations.” The Merriam-Webster dictionary proposes the following definition:

1: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

2: a branch of, operation in, or use of mathematics

Definitions like these are common, but don’t really convey a sense of what mathematics is. Saunders Mac Lane opened the first chapter of Mathematics: Form and Function (1986) with the following statement (p. 6):

Mathematics, at the beginning, is sometimes described as the science of Number and Space—better, of Number, Time, Space, and Motion.

A somewhat different idea of the scope of mathematics had already emerged before the start of the 20th century. A relatively well-known quote is Benjamin Pierce’s comment that mathematics is “the science that draws necessary conclusions” (Google reports more than 2,500 hits for this exact phrase). This is the opening sentence of his Linear Associative Algebra, published posthumously in 1881. He then expands on this conception of mathematics:

This definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective. Recent investigations, of which quaternions is the most noteworthy instance, make it manifest that the old definition is too restricted. The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law.

This conception of the scope of mathematics is much broader than one would expect from either the typical dictionary definitions or from the non-specialist’s experience of mathematics. For that matter, so is Mac Lane’s conception. In Mathematics: Form and Meaning, Mac Lane examines mathematics from several points of view, outlining and critiquing several schools of thought about its nature. Mac Lane evaluates the conception of mathematics as logicism, set theory, platonism, formalism, intuitionism, constructivism, finitism, and empiricism, all of which have been put forward as philosophical foundations for mathematics. He evaluates them, and finds all of them wanting (p. 456):

Each of these philosophies illuminates a relevant aspect of Mathematics, but none of them is remotely adequate as a description or foundation of the actual extensive network of Mathematics. Instead, our study has revealed Mathematics as an array of forms, codifying ideas extracted from human activities and scientific problems and deployed in a network of formal rules, formal definitions, formal axiom systems, explicit theorems with their careful proof and the manifold interconnections of these forms. More briefly, Mathematics aims to understand, to manipulate, to develop, and to apply those aspects of the universe which are formal.

The manipulation of numbers and geometric figures, and the establishment of their properties, certainly falls within the scope of mathematics as conceived of by these authors. However, mathematics is not limited to such considerations. Graph theory, for example, is unconcerned with the nature of the vertices of a graph. Graphs are used in modeling any set of (binary) relationships—whether that be shipping routes, network connections, inheritance relations in an object-oriented software system, dependencies in a project plan, or interspecies predator-prey relationships. Graph theory focuses on the existence of relationships between elements of a set, and systematically develops our understanding of the consequences that follow just from the existence of those relationships.

Eugene Wigner once wrote an article called The Unreasonable Effectiveness of Mathematics in the Natural Sciences. If we consider mathematics as a science of numbers, his assertion that mathematics is “unreasonably” effective appears to make sense. Once we discard this too-narrow idea of mathematics, the effectiveness of mathematics in scientific inquiry should no longer be a surprise. Mathematics is an effective tool in natural scientific inquiry precisely because the sciences are concerned with the systematic aspects of the phenomena they investigate. With this conception in hand, I suspect that the basis for judging mathematics to be unreasonably effective reduces to an a priori belief that the universe shouldn’t display any systematic properties.

Scientific inquiry into any set of phenomena presupposes that there is some degree of systematic behavior in what we observe. The scope of contemporary mathematics, as suggested by the perspectives offered above by Pierce and Mac Lane, can be considered as the systematic study of what it means to have a systematic behavior—in short, what Lynn Arthur Steen has called “the science of patterns”.