Object realism versus mathematical structuralism
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Krzysztof Wójtowicz,
Abstract
In this article, two possible standpoints concerning the nature of mathematical entities are discussed: object realism (according to which the identity of mathematical objects is determined by their intrinsic properties); and structuralism (according to which it is determined only by their relations to other mathematical objects). I discuss some motivations that underlie the structuralist view of mathematical entities, and indicate some philosophical drawbacks of this position. In particular, I argue, that these motivations and intuitions violate basic ontological intuitions concerning the notions of set and membership. I also claim that, from the point of view of mathematical structuralism, the main problem of object realism is the reference problem. On the other hand, from the point of view of object realism, mathematical structuralism has to accept very strong ontological assumptions (reifying classes of isomorphism). It is also not quite clear how the notion of identity is understood from the point of mathematical structuralism. Finally, I argue that the difference between mathematical structuralism and object realism is (at least partly) a matter of formulation.
© 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston
Articles in the same Issue
- Prelims
- Semiotics and logic: Pragmatization of the common ground
- Meaning between sense and reference: Impacts of semiotics on philosophy of science
- Where does logic meet semiotics?
- The correspondence theory of truth
- The intent to lie
- Reasoning in belief contexts
- Pragmatic constraints of meaning: An inferentialist approach
- On common knowledge in conversation
- Proofs and mistakes: Their syntactics, semantics, and pragmatics
- Object realism versus mathematical structuralism
- Indeterministic tense operators and the concept of time as a substance
- Counterfactual semantics and quantum physics
- The ultimate strengthening of the Turing Test?
- Is computation based on interpretation?
- The troubles with ontological primacy
- Some remarks on the word “be” and other existential expressions
- The evolution of scientific languages in Ajdukiewicz and Kuhn
- The core of grammar
- The grammar of philosophical discourse
- Semantic bounds for everyday language
- Demonstrative descriptions and conventional implicatures
- Does the Twin-Earth argument rest on a fallacy of equivocation?
Articles in the same Issue
- Prelims
- Semiotics and logic: Pragmatization of the common ground
- Meaning between sense and reference: Impacts of semiotics on philosophy of science
- Where does logic meet semiotics?
- The correspondence theory of truth
- The intent to lie
- Reasoning in belief contexts
- Pragmatic constraints of meaning: An inferentialist approach
- On common knowledge in conversation
- Proofs and mistakes: Their syntactics, semantics, and pragmatics
- Object realism versus mathematical structuralism
- Indeterministic tense operators and the concept of time as a substance
- Counterfactual semantics and quantum physics
- The ultimate strengthening of the Turing Test?
- Is computation based on interpretation?
- The troubles with ontological primacy
- Some remarks on the word “be” and other existential expressions
- The evolution of scientific languages in Ajdukiewicz and Kuhn
- The core of grammar
- The grammar of philosophical discourse
- Semantic bounds for everyday language
- Demonstrative descriptions and conventional implicatures
- Does the Twin-Earth argument rest on a fallacy of equivocation?
