Mathematics According to Structuralists

Structuralism

The structuralist holds that pure mathematics is the deductive study of structures. He believes that all mathematical theories describe structures, including non-algebraic ones. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. Mathematical knowledge consists in the ability to spot patterns.

Addressing Mathematical Objects

Structuralists believe that real objects of study in mathematics are structures or patterns. Numbers and other objects in math are just positions in a structure or pattern. To convey a clearer understanding, we will employ a baseball analogy: the shortstop position is one that can be filled with a lot of different players and is just a position. Each position is identified by certain regions on the baseball field. One way to think of this position is that shortstop is played by an actual person, Honus Wagner, and he is a specific object. Structuralists think of the shortstop position itself, it is not an object in this sense but rather a role that can be filled. This is similar to the number 4, for example. It represents the fourth position in the set of natural numbers, but different objects can be put into this position because the number itself is not an object at all, just a position.

Structuralists do not believe that numbers and other mathematical objects have internal properties. They only have their relation to other objects. Referring to the baseball analogy above, we know the shortstop position does not have a set height, weight, or batting average. It only has the structural properties of being located in or near the infield.

Addressing Structure

Define a pattern or structure of a system to be the abstract form of a structure, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system. To give an analogy, one can understand a basketball defense by going to a game and observing spatial relations and roles among players on the team without the ball, ignoring things like height, hair color, and field goal percentage.

So, the natural number structure is the form common to all of the natural number systems. And this structure is the subject matter of arithmetic. The Euclidean-space-structure is the form common to all Euclidean systems. The theme of structuralism is that, in general, the subject matter of a branch of mathematics is a given structure or a class of related structures—such as all algebraically closed fields.

Ante Rem and In Rebus

Ante remStructuralism (sometimes described as “structuralism without structures”)can be described in the following way: even if there were no infinite systems to be found in nature, the structure of the natural numbers would exist. In order to help the reader understand more fully, Shapiro uses an analogy: even if there were no red things in the world, the property of redness would still exist. Thus structures as Shapiro understands them are abstract, platonic entities.
According to In Rebusstructuralists, no abstract structures exist over and above the systems that instantiate them. Structures exist only in the systems that instantiate them. Continuing the analogy from above, he believes that if we were to get rid of all red things, then redness itself would go with them. In this sense, there is no more to redness than what all red things have in common.