Pictured to the left is Paul Benacerraf and on the right is David Hilbert.

Author: Graeham Rieman

###### September 24, 2014

Structuralism is, at its simplest, a viable alternative to Platonism the idea that
mathematical objects, just like any other objects, have Platonic forms. Instead, structuralism
posits that objects don't need to exist abstractly, or even have some inherent nature,
but that these objects gain their significance only from their relationships with
other mathematical objects. That is, 2 has no abstract existence, and no meaning beyond
that which it gains from its placement in the framework of the real numbers, being
more than 1, less than 3, and half of 4. This idea has been developed by many mathematic
philosophers, notably David Hilbert with his work Grundlagen der Geometrie (Foundations
of Geometry) and Paul Benacerraf with What Numbers Could Not Be and following papers,
and has developed into three distinct variants: Ante Rem, In Re, and Post Res.

Hilbert's 1899 work, Grundlagen der Geometrie, is a relatively early example of Structuralism
in practice. This work introduced the modern understanding of Euclidean geometry by
establishing a set of axioms that relate points, lines, and planes. He gives no explanation
of what points, lines, or planes intrinsically are a beyond providing notation to
refer to each and instead proceeds to define these three elements of geometry purely
by their relations to one another. He sets up five groups of axioms Axioms of Connection,
Axioms of Order, Axioms of Parallels, Axioms of Congruence, and Axioms of Continuity
wherein he defines the links between these objects. The Axioms of Connection are concerned
with relating these things to each other, defining lines and planes in terms of points,
and then points in terms of lines and planes. The Axioms of Order are concerned with
the understanding of "between-ness", and include the statement that "of any three
points situated on a straight line, there is always one and only one which lies between
the other two" (II, 3).

From these two sections, he establishes theorems concerning the properties of points
on a line that, between any two points on a line, there are an unlimited number of
points and the properties and definition of line segments, which lie between two
points on a plane that are each on different sides of a line that is also in the plane.
The last three sections define parallels, congruence, and continuity, again using
points, lines, and planes, as well as the previously given axioms. In establishing
his geometry in this way, Hilbert's axioms the statements that relate the three distinct
systems of points, lines, and planes are what are established as true statements,
with no concern given to what an individual point, line, or plane is in and of itself.
In doing so, he rejects Platonism's idea of the existence of mathematical objects
in favor of a truth comprised of a set of relationships, but the focus of the work
was on the mathematics itself rather than the philosophy behind it.

Sixty-six years later, Paul Benacerraf's What Numbers Could Not Be followed a philosophical
path, beginning with the consideration of a unique style of math education, one that
consisted of learning logic and set theory before encountering numbers at all. In
this example, the student, whom he names Ernie, is exposed to the natural numbers
after learning about sets, and is able to easily correspond these entities to the
elements of the infinite set that Beneceraf denotes as , with an ordering R that corresponds
to the less-than of the natural numbers. With this correlation, Ernie could then prove
all the properties of the natural numbers using only his understanding of this set
.

Further, the operations on the natural numbers, e.g. addition, multiplication, and
exponentiation, he would easily understand as operations on the members of rather
than having to define them recursively, as Benecreaf claims that an ordinary numbers-based
education requires one to do. Lastly, Ernie would have to grasp the application of
these newfound numbers, namely in counting. Beneceraf postulates that Ernie would
count transitively, an act which he defines as "learning their [these numbers'] use
as measures of sets," (p. 274) which would consist of comparing the cardinality of
sets to with a one-to-one correspondence. In this way, Ernie would come to a full
understanding of numbers and their properties and uses without ever addressing what
a number is, but instead using the framework, or structure, that he already knew to
give them meaning. As such, Ernie is a perfect example of the ability of structuralism
to explain the operations of mathematics.

From these works, and other, later works, structuralism has developed into three branches,
which span the philosophical gap between Platonism and Anti-Realistic understandings.
The most Platonistic of these views, the Ante Rem rendering, holds that the structures
that give meaning to numbers are real in and of themselves, while the numbers are
not. The structures are assumed to have their own abstract form and existence, in
the same way that the Platonic Forms do. It differs from Platonism in that it rejects
the abstract existence of numbers, but is not different enough to avoid the problems
Platonism raises, namely the inability to prove or disprove Plato's theory of forms.

The other two versions reject Platonism's idea of abstract existence entirely. The
first of these, In Re, is a somewhat Aristotelian understanding that holds that structures
exist abstractly, but only to the extent that they are reflected or exemplified in
some system. However, this understanding assumes that the system comes into being
from its concrete example, and so it fails to deem some structures as truth by this
limitation alone. For example, the set of real numbers though it is widely accepted
in mathematics, and Ernie could come to an understanding of it in Beneceraf's thought
experiment would not be seen as a valid system with abstract existence in this understanding,
because it has no perfectly corresponding example that has been found in observable
reality.

The third and final variant, Post Res, is much closer to Nominalism. It holds mathematical
structures as a tool that can be used to explain the properties of the members, while
neither the structures nor the members necessarily exist in the abstract. Mathematical
structures, then, can only be considered in regard to the systems which reflect them.
The obvious weakness of this understanding is that these structures themselves are
then much less useful and cannot always be used to prove truths about their members.

Thus, Structuralism provides a viable alternative to Platonism and Nominalism, but
cannot entirely escape the difficulties and problems posed by either. It is, however,
an extremely useful understanding for abstract or applied mathematics, since it holds
the system and its rules as the truth that can give meaning and explanation to any
entity that fits into it, even if that entity itself is difficult to understand.

*Works Cited*

*Benacerraf, Paul. "What numbers could not be", Philosophy of Mathematics. 2nd ed.
Cambridge: Cambridge University Press, 1984. 272-294. Cambridge Books Online. Web.
07 March 2014. <http://dx.doi.org/10.1017/CBO9781139171519.015>.*

*Hilbert, David (1950) [first published 1902], The Foundations of Geometry [Grundlagen
der Geometrie], English translation by E.J. Townsend (2nd ed.), La Salle, IL: Open
Court Publishing. Web. 07 March 2014. <http://www.gutenberg.org/files/17384/17384-pdf.pdf>.*