formalism the view that mathematics concerns manipulations of symbols according to prescribed structural rules. It is cousin to nominalism, the older and more general metaphysical view that denies the existence of all abstract objects and is often contrasted with Platonism, which takes mathematics to be the study of a special class of non-linguistic, non-mental objects, and intuitionism, which takes it to be the study of certain mental constructions. In sophisticated versions, mathematical activity can comprise the study of possible formal manipulations within a system as well as the manipulations themselves, and the ‘symbols’ need not be regarded as either linguistic or concrete. Formalism is often associated with the mathematician David Hilbert. But Hilbert held that the ‘finitary’ part of mathematics, including, for example, simple truths of arithmetic, describes indubitable facts about real objects and that the ‘ideal’ objects that feature elsewhere in mathematics are introduced to facilitate research about the real objects. Hilbert’s formalism is the view that the foundations of mathematics can be secured by proving the consistency of formal systems to which mathematical theories are reduced. Gödel’s two incompleteness theorems establish important limitations on the success of such a project. See also ABSTRACT ENTITY , AES- THETIC FORMALISM , HILBERT ‘s PROGRAM , MATHEMATICAL INTUITIONISM , PHILOSOPHY OF MATHEMATIC. S.T.K.