A nominalist says '7 is prime' is true without there being a number 7. A Platonist asks: 'What makes the statement true, if not a real number?' Which nominalist strategy answers by treating mathematical claims as true within a representational fiction?
AStructuralism — which holds that only relational patterns, not specific objects, are what mathematics is about
BNominalization programs — which rewrite mathematical claims as claims about concrete physical quantities
CFictionalism — which says mathematical claims are true-in-the-mathematical-fiction, as 'Holmes lives at 221B Baker Street' is true-in-Conan-Doyle's-fiction, without a real referent
DEliminativism — which denies that mathematical claims have truth values at all
Fictionalism (associated with Hartry Field) holds that mathematical discourse is indispensable as a representational tool without being literally true. '7 is prime' is true in the mathematical fiction — the same logical relation Holmes bears to Baker Street. This avoids positing abstract objects while explaining why mathematical claims have the logical structure they do. The challenge is then explaining why this 'fiction' is so reliable for real scientific work.
Question 2 Multiple Choice
The Benacerraf challenge is often called an epistemological objection to Platonism. What is the core worry?
APlatonism cannot explain why different cultures develop different mathematical systems
BIf abstract objects exist outside space and time and are causally inert, it is unclear how humans — physical beings embedded in causal networks — could come to know truths about them
CPlatonism requires mathematical knowledge to be infallible, which conflicts with the historical record of mathematical errors
DPlatonism posits too many abstract objects, violating Ockham's razor more than any alternative view
Benacerraf's challenge (1973) highlights a tension between two plausible requirements: mathematical statements are objectively true, and knowledge requires some causal connection between knower and known. If numbers are causally inert and outside spacetime, no such connection seems possible — yet we seem to have mathematical knowledge. This epistemological pressure is part of what motivates nominalist alternatives even for philosophers who find Platonism otherwise attractive.
Question 3 True / False
Fictionalism is the most successful nominalist strategy because it fully resolves the problem of explaining why mathematical fiction is reliable enough to do actual scientific work.
TTrue
FFalse
Answer: False
This is precisely the challenge fictionalism has not fully answered. If mathematics is a fiction, what accounts for its extraordinary precision and predictive power in physics and engineering? A Sherlock Holmes story does not predict real criminal behavior, but mathematical fictions apparently do predict the behavior of physical systems. Fictionalists must explain this without re-introducing abstract objects through the back door, and no consensus solution exists.
Question 4 True / False
Nominalism's primary motivation is ontological parsimony — the principle that we should not posit more kinds of entities than are needed to explain what needs explaining.
TTrue
FFalse
Answer: True
Ockham's razor applied to ontology: if mathematical and scientific discourse can be made sense of without positing an abstract realm of numbers, sets, and propositions, then we have no reason to believe such a realm exists. The nominalist's first move is always to show that the explanatory work Platonists assign to abstract objects can be done without them — or at least to challenge the assumption that Platonism's explanatory gains outweigh its ontological costs.
Question 5 Short Answer
Each major nominalist strategy trades one problem for another. Briefly identify the main challenge facing fictionalism, structuralism, and nominalization programs, and explain why the debate between nominalism and Platonism remains unresolved.
Think about your answer, then reveal below.
Model answer: Fictionalism must explain why a mere fiction is reliable enough to do real scientific work. Structuralism must explain what structures are without reintroducing abstract objects as their constituents. Nominalization programs have proved difficult to complete for all of physics. The debate persists because Platonism faces Benacerraf's epistemological challenge (how do we know about causally inert abstracta?), so neither side has a cost-free theory.
The nominalism-Platonism debate is a genuine theoretical impasse: Platonism explains mathematical truth cleanly but struggles with epistemology; nominalism has good epistemological credentials but cannot yet fully reconstruct mathematical discourse without abstract objects. Progress requires either a compelling nominalization of all of mathematics and science, or a Platonist epistemology that explains abstract knowledge without causal contact — and neither has been achieved.