Concept
Structuralism (philosophy of mathematics)
Overview
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. 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 numbers; and, by generalization of this example, that any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include lines 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 concerns the kind of entity a mathematical object is, rather than the kind of existence mathematical objects or structures have (i.e., their ontology). The kind of existence that mathematical objects have would depend 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, Geoffrey Hellman, Michael Resnik, Stewart Shapiro and James Franklin.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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The logical basis of the theory alone belongs to philosophy, and alone will occupy us to-night. The way the problem of continuity enters into philosophy is, broadly speaking, the following: Space and time are treated by mathematicians as consisting of points and instants, but they also have a property, easier to feel than to define, which is called continuity, and is thought by many philosophers to be destroyed when…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Structuralism is.
For when we regard a quantity as extensive, _i.e._ as divisible into parts, we necessarily regard it as more than a mere relation or adjective, since no mere relation or adjective can be divided. For quantitative treatment, therefore, spatial relations must be hypostatized[140]. When this is done, we obtain, as we saw above, a homogeneous and infinitely divisible form of externality.
Bertrand Russell, An essay on the foundations of geometry
States what Structuralism is.
Meanwhile every mathematical proposition remains always a universal truth, which is valid for innumerable particular cases; and a graduated process from the simple to the complicated propositions which are to be deduced from them, is also essential to mathematics; therefore, in every respect mathematics is a science. The completeness of a science as such, that is, in respect of form, consists in there being as much…
Arthur Schopenhauer, The World as Will and Idea
States what Structuralism is.
Philosophers who discussed Structuralism (philosophy of mathematics)
Related ideas
- Philosophical realism
- Paul Benacerraf
- Truth value
- Function (mathematics)
- Geometry
- Mathematical object
- Intuitionism
- Real number
- Nominalism
- Natural number
- Anti-realism
- Platonism
- Ontology
- Edward N. Zalta
- Medieval philosophy
- Set theory
- Mathematical Platonism
- Crispin Wright
- Philosophy of mathematics
- Abstract and concrete
- System
- Epistemology
- Bob Hale (philosopher)
- Aristotelian realist philosophy of mathematics
Place in the tradition
Part of Analytic philosophy.
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