Structuralism (philosophy of mathematics)
Encyclopedia
Structuralism is a theory in the philosophy of mathematics
that holds that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their place in such structures, consequently having no intrinsic properties
. For instance, it would maintain that all that needs to be known about the number 1 is that is its the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line
. Other examples of mathematical objects might include line
s and planes in geometry, or elements and operations in abstract algebra
.
Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what kind of entity a mathematical
object is, not to what kind of existence mathematical objects or structures have (not, in other words,
to their ontology
). The kind of existence mathematical objects have would clearly be dependent on that of the
structures in which they are embedded; different sub-varieties of structuralism make different ontological claims
in this regard.
The Ante Rem ("before the thing"), or fully realist, variation of structuralism has a similar ontology to Platonism in that structures are held to have a real but abstract and immaterial existence. As such, it faces the usual problems of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
In Re ("in the thing"), or moderately realistic, structuralism is the equivalent of Aristotelean realism. Structures are held to exist
inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly
legitimate structures might accidentally happen not to exist, and that a finite physical world might
not be "big" enough to accommodate some otherwise legitimate structures.
The Post Res ("after things") or eliminative variant of structuralism is anti-realist
about structures in a way that parallels nominalism
. According to this view mathematical systems exist, and have structural features
in common. If something is true of a structure, it will be true of all systems exemplifying the structure.
However, it is merely convenient to talk of structures being "held in common" between systems: they in fact have no independent existence.
Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf
, Michael Resnik
and Stewart Shapiro
.
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
that holds that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their place in such structures, consequently having no intrinsic properties
Intrinsic and extrinsic properties (philosophy)
An intrinsic property is a property that an object or a thing has of itself, independently of other things, including its context. An extrinsic property is a property that depends on a thing's relationship with other things...
. For instance, it would maintain that all that needs to be known about the number 1 is that is its the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line
Number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...
. Other examples of mathematical objects might include line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s and planes in geometry, or elements and operations in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
.
Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what kind of entity a mathematical
object is, not to what kind of existence mathematical objects or structures have (not, in other words,
to their ontology
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
). The kind of existence mathematical objects have would clearly be dependent on that of the
structures in which they are embedded; different sub-varieties of structuralism make different ontological claims
in this regard.
The Ante Rem ("before the thing"), or fully realist, variation of structuralism has a similar ontology to Platonism in that structures are held to have a real but abstract and immaterial existence. As such, it faces the usual problems of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
In Re ("in the thing"), or moderately realistic, structuralism is the equivalent of Aristotelean realism. Structures are held to exist
inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly
legitimate structures might accidentally happen not to exist, and that a finite physical world might
not be "big" enough to accommodate some otherwise legitimate structures.
The Post Res ("after things") or eliminative variant of structuralism is anti-realist
Anti-realism
In analytic philosophy, the term anti-realism is used to describe any position involving either the denial of an objective reality of entities of a certain type or the denial that verification-transcendent statements about a type of entity are either true or false...
about structures in a way that parallels nominalism
Nominalism
Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist. Thus, there are at least two main versions of nominalism...
. According to this view mathematical systems exist, and have structural features
in common. If something is true of a structure, it will be true of all systems exemplifying the structure.
However, it is merely convenient to talk of structures being "held in common" between systems: they in fact have no independent existence.
Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf
Paul Benacerraf
Paul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....
, Michael Resnik
Michael Resnik
Michael David Resnik is a leading contemporary philosopher of mathematics. He obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his Ph.D. in Philosophy at Harvard University in 1964. He wrote his thesis on Frege...
and Stewart Shapiro
Stewart Shapiro
Stewart Shapiro is O'Donnell Professor of Philosophy at the Ohio State University and a regular visiting professor at the University of St Andrews in Scotland. He is an important contemporary figure in the philosophy of mathematics where he defends a version of structuralism. He studied...
.
Further reading
- RESNIK, Michael (1997), Mathematics as a Science of Patterns, Clarendon Press, Oxford, UK, ISBN 9780198250142
- SHAPIRO, Stewart (2000), Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, UK, ISBN 0-19-289306-8
External links
- Mathematical Structuralism, Internet Encyclopaedia of Philosophy
- Foundations of Structuralism research project, University of Bristol, UK