Abstract Objects
Abstract objects are entities such as numbers, sets, properties, and propositions that, if they exist, are non-spatial, non-temporal, and causally inert. This topic examines whether such objects exist and how we could know about them.
Definition
An abstract object is an entity lacking spatial location and causal powers, contrasted with concrete objects located in space and time.
Scope
Covers the abstract-concrete distinction, platonism versus nominalism, the indispensability argument for mathematical objects, Field's nominalist program, and the epistemological problem of access to causally inert entities.
Core questions
- Do abstract objects such as numbers and properties exist?
- How should the abstract-concrete distinction be drawn?
- If abstracta are causally inert, how could we know about them?
- Can mathematics be done without quantifying over abstract objects?
Key concepts
- Abstract-concrete distinction
- Platonism
- Nominalism
- Causal inertness
- Indispensability argument
Key theories
- Platonism about abstracta
- Numbers, sets, and other abstracta exist as mind-independent, non-spatiotemporal entities; mathematical statements are literally true and refer to them.
- Nominalism
- There are no abstract objects; apparent reference to them is to be paraphrased away or treated instrumentally, as in Field's program of doing science without numbers.
History
Plato's Forms are the ancestral abstract objects. Frege defended numbers as objective, non-physical entities. The Quine-Putnam indispensability argument tied mathematical platonism to scientific realism, while Benacerraf sharpened the epistemological challenge and Field developed a systematic nominalism in reply.
Debates
- The epistemological access problem
- Benacerraf argued that if abstracta are causally isolated, the best theory of knowledge cannot explain how we know mathematical truths; platonists and nominalists respond in opposing ways.
Key figures
- Gottlob Frege
- Willard Van Orman Quine
- Paul Benacerraf
- Hartry Field
- Gideon Rosen
Related topics
Seminal works
- frege1884
- field1980
- benacerraf1973
Frequently asked questions
- What makes an object abstract rather than concrete?
- The most common criterion is that abstract objects lack spatial and temporal location and stand in no causal relations, unlike concrete objects such as tables or electrons. The exact criterion is itself debated.