In addition, mathematics has a metalanguage; that is, names for mathematical statements and other parts of syntax, self-reference, proof and truth.
As will become clear below, it is common that nominalization strategies for mathematics change either the syntax or the semantics of mathematical statements.
The nominalistic account of mathematics that will now be discussed holds that all existential mathematical statements are false simply because there are no mathematical entities.
For example we can consider misinterpretations I of a fully interpreted sentence S. A misinterpretation of S that makes it true is known as a nonstandard or unintended model of S. The branch of mathematics called nonstandard analysis is based on nonstandard models of mathematical statements about the real or complex number systems; see Section 4 below.
Philosophical views concerning the ontology of mathematics run the gamut from platonism (mathematics is about a realm of abstract objects), to fictionalism (mathematics is a fiction whose subject matter does not exist), to formalism (mathematical statements are meaningless strings manipulated according to formal rules), with no consensus about which is correct.
And Ludwig Wittgenstein, in his Remarks on the Foundations of Mathematics (1956), can be seen as recommending a noncognitivist approach to logical and mathematical statements, according to which they do not record truths of some special kind but rather express rules which regulate the use of more ordinary or empirical statements.
This means two things: mathematical statements are general, and they are confirmed by proof.
A mathematician friend recently explained that there is a whole class of mathematical statements that can be called independent.
Similar remarks can be made about the other versions of paraphrase nominalism; all of these views involve the same idea that mathematical statements are not used literally.
Classical mathematicians also accept assumption 2 and therefore reluctantly agree with Gödel that, contrary to Hilbert's expectation, there are true mathematical statements which are not provable.
The mathematical statements that can be proved are called theorems, and it follows that, in principle, a mechanical device, such as a modern computer, can generate all theorems.
