A student argues that numbers must exist because '7 is prime even if no one had ever thought about it.' A nominalist responds that this doesn't require an object called 'seven' to exist robustly. Which nominalist strategy does this response exemplify?
AThe indispensability argument — accepting that mathematics is indispensable to science and therefore abstract objects exist
BFictionalism — mathematical statements can be systematically true within a useful fiction without requiring that the entities they mention actually exist
CModal structuralism — rephrasing mathematical claims as claims about what structures would exist if certain axioms held
DTrope theory — replacing abstract universal numbers with concrete particular instances of numerosity
Fictionalism holds that mathematical statements are 'true within a fiction' — like 'Sherlock Holmes lives at 221B Baker Street,' which is true within Doyle's fiction without implying a real detective. Similarly, '7 is prime' is systematically true within the mathematical fiction without requiring a Platonic object 'seven' to exist. This preserves the utility and inferential structure of mathematics while rejecting its face-value ontological commitment.
Question 2 Multiple Choice
The 'epistemological problem' for Platonism about abstract objects is that:
AAbstract objects are too vaguely defined to distinguish them precisely from physical objects
BIf abstract objects are non-spatial, non-temporal, and causally inert, it is unclear how cognitive faculties that evolved to track physical environments through causal contact could yield knowledge about them
CAccepting abstract objects conflicts with modern physics, which leaves no room for non-physical entities
DAbstract objects would make mathematical truths too certain, eliminating the possibility of genuine mathematical discovery
Our knowledge-forming faculties work through causal chains: perception, memory, and inference are all causally mediated. But abstract objects — non-spatial, non-temporal, and causally inert — cannot enter any causal chain. The Platonist must explain how finite, physical minds come to know a realm that by definition cannot affect them. The standard responses — positing special rational intuition, or revising the causal theory of knowledge — each carry significant philosophical costs.
Question 3 True / False
The Quine-Putnam indispensability argument concludes that we have reason to believe abstract objects exist because our best scientific theories quantify over mathematical entities and we believe those theories are true.
TTrue
FFalse
Answer: True
The argument runs: (1) we are committed to the existence of entities our best theories quantify over; (2) our best scientific theories quantify over mathematical objects (sets, functions, numbers); (3) therefore we are committed to the existence of mathematical objects. This grounds abstract objects empirically by tying them to theoretical commitments we already accept when we accept physics, rather than grounding them in pure a priori Platonic intuition.
Question 4 True / False
Nominalists should deny that mathematical statements like '7 is prime' are true, since on their view there is no object '7' for the statement to be about.
TTrue
FFalse
Answer: False
This is a common mischaracterization. Most nominalist strategies preserve mathematical truth without ontological commitment to abstract objects. Fictionalists accept that '7 is prime' is true within a mathematical fiction. Modal structuralists paraphrase it as a claim about what would hold in any Peano-satisfying structure. Neither strategy declares mathematics false — they provide alternative analyses of what mathematical truth amounts to. Nominalism is a position about ontology, not about mathematical correctness.
Question 5 Short Answer
What is the epistemological problem for Platonism, and why does it represent a genuine philosophical challenge rather than a merely verbal puzzle?
Think about your answer, then reveal below.
Model answer: Platonism holds that abstract objects exist in a non-spatial, non-temporal realm, causally isolated from the physical world. Our cognitive faculties — perception, memory, inference — evolved to track physical environments through causal interaction, and knowledge through perception requires that information causally reach us. Since abstract objects are causally inert, no such contact is possible. Every proposed solution has costs: positing 'rational intuition' is mysterious; revising the causal theory of knowledge undermines broader epistemology; fictionalism preserves utility but gives up the view that mathematics is about anything real.
The problem sits at the intersection of metaphysics and epistemology. You can't assess abstract-objects ontology by asking only what makes mathematical discourse true — you must also ask whether proposed entities are epistemically accessible in a way compatible with how minds work. Platonism's strength is explaining mathematical necessity, objectivity, and applicability; its weakness is explaining cognition. This trade-off drives the ongoing debate between Platonist and nominalist positions in philosophy of mathematics.