Structuralism

Structuralism is a general approach in various academic disciplines that seeks to explore the inter-relationships between some fundamental elements, upon which higher mental, linguistic, social, cultural etc "structures" are built, through which then meaning is produced within a particular person, system, culture.

Structuralism in the Philosophy of Mathematics

Structuralism in mathematics is the study of what structures say a mathematical object is, and how the ontology of these structures should be understood. This is a growing philosophy within mathematics that is not without its share of critics.

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In 1965, Paul Benacerraf wrote a paper entitled: "What Numbers Could Not Be." This paper is a seminal paper on mathematical structuralism in an odd sort of way: it started the movement by the response it generated. Benacerraf addressed a notion in mathematics to treat mathematical statements at face value, in which case we are committed to a world of an abstract, eternal realm of mathematical objects. Bernacerraf's dilemma is how do we come to know these objects if we do not stand in causal relation to them. These objects are considered causally inert to the world. Another problem raised by Bernacerraf is the multiple set theories that exist by which reduction of elementary number theory to sets is possible. Deciding which set theory is true has not been feasible. Benacerraf concluded in 1965 that numbers are not objects.

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The answer to Benacerraf's negative claims is how structuralism became a viable philosophical program within mathematics. The structuralist responds to these negative claims that the essence of mathematical objects is relations that the objects bear with the structure. Structures are exemplified in abstract systems in terms of the relations that hold true for that system.

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Important contributions to structuralism in Mathematics have been made by Bourbaki.

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