The assumption that continuous mathematics maps exactly onto physical reality commits a category error identified by Hartry Field and structural realists: mathematical representation is not identity with the target domain.

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Category error(as used in logic and philosophy of language): A logical mistake where you apply a rule or concept to something it doesn't actually fit, like using a math formula on a poem.
Continuous mathematics(as used in philosophy of mathematics): A type of math that deals with smooth, unbroken quantities (like real numbers) rather than separate, distinct units—think of it as the math of flowing curves rather than individual dots.
Hartry Field(the author of the nominalist program discussed): A contemporary philosopher who developed a theory showing that we might not actually need to believe in abstract objects (like numbers) even though they seem useful in science and math.
Mathematical representation(as used in philosophy of science): Using math (equations, numbers, formulas) to describe or model something in the real world—like using an equation to describe how a ball falls.
structural realism(Philosophy of science; the position all three critics are attacking): The view that our best scientific theories give us knowledge of the structure of the world but not of its intrinsic nature or content, typically motivated by the problem of theory change
target domain(Formal structure of analogical arguments): The domain T about which an inference is being made by analogy with the source domain; has the same formal structure as the source domain.

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