Structuralism is a theory in the
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in peopl ...
that holds that mathematical theories describe structures of
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
s. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any
intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. By generalization of this example, any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
s and
planes
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes'' ...
in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, or elements and
operations
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
.
Structuralism is an
epistemologically realistic view in that it holds that mathematical statements have an objective
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false'').
Computing
In some prog ...
. 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
In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality.
Ontology addresses questions like how entities are grouped into categories and which of these entities ...
). The kind of existence that mathematical objects have would be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard.
Structuralism in the philosophy of mathematics is particularly associated with
Paul Benacerraf
Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He wa ...
,
Geoffrey Hellman,
Michael Resnik
Michael David Resnik (; born March 20, 1938) is a leading contemporary American philosopher of mathematics.
Biography
Resnik obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his PhD in Philosophy at Harvard Univ ...
,
Stewart Shapiro and
James Franklin.
Historical motivation
The historical motivation for the development of structuralism derives from a fundamental problem of
ontology
In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality.
Ontology addresses questions like how entities are grouped into categories and which of these entities ...
. Since
Medieval
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
times, philosophers have argued as to whether the ontology of mathematics contains
abstract object
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
s. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that: (1) exists independent of the mind; (2) exists independent of the empirical world; and (3) has eternal, unchangeable properties. Traditional mathematical
Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at ...
maintains that some set of mathematical elements–
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s,
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s,
functions,
relations
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
,
systems–are such abstract objects. Contrarily, mathematical
nominalism
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings th ...
denies the existence of any such abstract objects in the ontology of mathematics.
In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included
intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of ...
,
formalism
Formalism may refer to:
* Form (disambiguation)
* Formal (disambiguation)
* Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
* Formalism (linguistics)
* Scien ...
, and
predicativism. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for structuralism developed. In 1965,
Paul Benacerraf
Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He wa ...
published a paradigm changing article entitled "What Numbers Could Not Be".
Benacerraf concluded, on two principal arguments, that
set-theoretic
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
Platonism cannot succeed as a philosophical theory of mathematics.
Firstly, Benacerraf argued that Platonic approaches do not pass the ontological test.
He developed an argument against the ontology of set-theoretic Platonism, which is now historically referred to as
Benacerraf's identification problem. Benacerraf noted that there are
elementarily equivalent In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one ofte ...
, set-theoretic ways of relating natural numbers to
pure sets In set theory, a hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on.
Examples
For example, it is vacuously true that ...
. However, if someone asks for the "true" identity statements for relating natural numbers to pure sets, then different set-theoretic methods yield contradictory identity statements when these elementarily equivalent sets are related together.
This generates a set-theoretic falsehood. Consequently, Benacerraf inferred that this set-theoretic falsehood demonstrates it is impossible for there to be any Platonic method of reducing numbers to sets that reveals any abstract objects.
Secondly, Benacerraf argued that Platonic approaches do not pass the
epistemological
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
test. Benacerraf contended that there does not exist an empirical or rational method for accessing abstract objects. If mathematical objects are not spatial or temporal, then Benacerraf infers that such objects are not accessible through the
causal theory of knowledge.
The fundamental epistemological problem thus arises for the Platonist to offer a plausible account of how a mathematician with a limited, empirical mind is capable of accurately accessing mind-independent, world-independent, eternal truths. It was from these considerations, the ontological argument and the epistemological argument, that Benacerraf's anti-Platonic critiques motivated the development of structuralism in the philosophy of mathematics.
Varieties
Stewart Shapiro divides structuralism into three major schools of thought.
These schools are referred to as the ''ante rem'', the ''in re'', and the ''post rem''.
The ''ante rem'' structuralism
("before the thing"), or abstract structuralism
or abstractionism (particularly associated with
Michael Resnik
Michael David Resnik (; born March 20, 1938) is a leading contemporary American philosopher of mathematics.
Biography
Resnik obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his PhD in Philosophy at Harvard Univ ...
,
Stewart Shapiro,
Edward N. Zalta, and
Øystein Linnebo) has a similar ontology to
Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at ...
(see also
modal neo-logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or al ...
). Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem, as noted by Benacerraf, of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
The ''in re'' structuralism
("in the thing"),
or modal structuralism
(particularly associated with
Geoffrey Hellman),
is the equivalent of
Aristotelian realism
Aristotle's Theory of Universals is Aristotle's classical solution to the Problem of Universals, sometimes known as the hylomorphic theory of immanent realism. Universals are the characteristics or qualities that ordinary objects or things ha ...
(realism in truth value, but
anti-realism
In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument ...
about
abstract object
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
s in ontology). 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 Aristotelian realism of James Franklin is also an ''in re'' structuralism, arguing that structural properties such as symmetry are instantiated in the physical world and are perceivable.
In reply to the problem of uninstantiated structures that are too big to fit into the physical world, Franklin replies that other sciences can also deal with uninstantiated universals; for example the science of color can deal with a shade of blue that happens not to occur on any real object.
The ''post rem'' structuralism
("after the thing"), or eliminative structuralism
(particularly associated with
Paul Benacerraf
Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He wa ...
),
is
anti-realist
In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument ...
about structures in a way that parallels
nominalism
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings th ...
. Like nominalism, the ''post rem'' approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. 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 instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.
See also
*
Abstract object theory
Abstract object theory (AOT) is a branch of metaphysics regarding abstract objects. Originally devised by metaphysician Edward Zalta in 1981, the theory was an expansion of mathematical Platonism.
Overview
''Abstract Objects: An Introduction ...
*
Foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
*
Univalent foundations
*
Aristotelian realist philosophy of mathematics
In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world (or in any other world there might be). It contrasts wi ...
;Precursors
*
Nicolas Bourbaki
References
Bibliography
*
*
*
*
External links
''Mathematical Structuralism'', Internet Encyclopaedia of Philosophy''Abstractionism'', Internet Encyclopaedia of PhilosophyFoundations of Structuralism research project University of Bristol, UK
{{Foundations-footer
History of mathematics
Mathematical logic
Philosophy of mathematics
Set theory
Abstract object theory