Quine and Putnam's indispensability arguments ground mathematical ontology in the explanatory and predictive success of scientific theories, not in the mere formal consistency or categoricity of logical systems.
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Mathematical ontology(what the indispensability argument is trying to ground or justify)
The question of what kinds of mathematical objects exist (like numbers, sets, or equations) and whether they're real or just useful inventions.
Ontology(Carnap argues this enterprise is based on a mistake)
The philosophical discipline that tries to answer hard questions about what there really is.
Putnam
# Putnam
"Putnam" most commonly refers to **Hilary Putnam** (1926-2016), an influential American philosopher who made major contributions to philosophy of mind, language, and science. He is famous for thought experiments like the "brain in a vat" scenario, which explores questions about reality and how we know what's real. His work fundamentally changed how philosophers think about the relationship between our minds, language, and the external world.
Quine(as a proper name referring to the philosopher whose theory is being discussed)
Willard Van Orman Quine was a 20th-century American philosopher who wrote about how we know things and how language works. In this statement, we're discussing one of his specific ideas about observation.
categoricity(Joyce's term for the inescapable practical force of moral demands)
The property of moral requirements whereby they apply to agents unconditionally, regardless of the agent's contingent desires, goals, or interests