While the Quine-Putnam indispensability argument posits that we must commit to the ontological reality of mathematical entities because they are inextricably bound up with our best scientific theories, nominalists such as Hartry Field have sought to demonstrate that science can be 'nominalized'—that is, reformulated without reference to abstract mathematical objects. Field's ambitious project involved rewriting Newtonian gravitational theory using only space-time points and regions. Critics of Field, however, argue that this approach relies on a substantivalist view of space-time, which itself postulates entities just as ontologically problematic as the numbers and sets the nominalist seeks to avoid. Furthermore, as David Malament has pointed out, it remains highly doubtful whether more complex scientific frameworks, such as quantum mechanics—which relies essentially on infinite-dimensional Hilbert spaces—can be subjected to a similar nominalist translation.
In the context of the passage, the author's reference to 'infinite-dimensional Hilbert spaces' serves primarily to:
- identify a specific theoretical feature that purportedly renders a modern scientific framework intractable to a proposed philosophical reformulationCevap
- Belucidate the precise mathematical constructs that the indispensability argument posits as fundamentally necessary for all scientific inquiry
- Cdemonstrate that the postulation of abstract entities is logically inescapable within any rigorously formalized physical theory
- Dsuggest that classical frameworks like Newtonian mechanics lack the requisite complexity to serve as adequate test cases for nominalism
- Ehighlight a critic's fundamental skepticism regarding the scientific validity of frameworks that rely heavily on abstract mathematical objects