Card 1.1.4 · Root I — Metaphysics
Abstract Objects
Does the number seven exist?
This node asks whether things like numbers, sets, propositions and properties exist, and if so what kind of existence that could be.
Where it sits
Under 1.1 Ontology. Borders 6.9 Philosophy of mathematics, 1.1.2 Universals, 2.2.2 Reason and intuition.
The rival answers
| Position | Abstract objects | The cost |
|---|---|---|
| Platonism | Exist, outside space and time | Benacerraf: it is unclear how we could know causally inert objects — one horn of his dilemma between truth and knowledge |
| Nominalism | Do not exist; talk of them is paraphrasable | The paraphrases get very hard for mathematics |
| Fictionalism | Do not exist; the talk is useful fiction | Indispensability: science quantifies over them |
| Conceptualism | Exist as mental constructions | Implies none existed before there were minds |
| Trope theory | Only particular property-instances exist | Needs primitive exact resemblance |
| Deflationism | The existence question is not substantive | Inherits the difficulties discussed at 1.10 |
Key figures
| Figure | Dates | Tradition | Contribution |
|---|---|---|---|
| Plato | 428–348 BCE | Greek | Forms as separate intelligible objects; particulars participate in them |
| Gottlob Frege | 1848–1925 | Analytic | Numbers as objective and non-physical; logicism |
| Paul Benacerraf | 1931–2025 | Analytic | The dilemma between semantics and epistemology |
| W. V. O. Quine | 1908–2000 | Analytic | Ontological commitment; the indispensability argument |
| Hilary Putnam | 1926–2016 | Analytic | An indispensability argument for mathematical objectivity; in later work, Putnam denied that it was intended as an argument for Platonism |
| Hartry Field | b. 1946 | Analytic | Science Without Numbers; nominalism attempted technically |
Connections
→ 1.1 · ↔ 1.1.2, 6.9, 2.2.2, 1.10 · ◇ platonism, nominalism, fictionalism, conceptualism, trope theory, deflationism · ○ Axial Age; Modern; Table 1
→ narrower questions · ↔ questions it borders · ◇ the positions, mapped in Part III of the book · ○ the eras and comparison tables (Parts IV and V)
The full card is in the book
In The Coordinates of Thought this card goes on to a worked example that sets the answers against one case, and to where the question touches daily life — page 32.
On Coordinates of Thought
Conversations about this question in Philosophy of mathematics:
- Are the words of the Veda abstract objects? Mīmāṃsā’s eternal language and Frege’s third realm 6 replies
- Reading group: Frege’s Foundations of Arithmetic (1884), five weeks with Austin’s translation 6 replies
- The actual infinite is a useful fiction, not something real 5 replies
- Numbers exist independently of us: mathematical platonism is true 5 replies
- My model has ‘the average borrower’. Nobody is the average borrower. What am I talking about? 5 replies
Join free to ask about it Everything in Philosophy of mathematics