At a recent conference held just inside Maubeuge's walls, the two worlds met for three days of abstract curves, weird shapes on screens and, of course, equations on blackboards.For, given any mathematical statement, a shape that the eye can savour is never far away.
For even if there are mathematical objects, referential and quantificational indeterminacy may deprive mathematical statements of a unique and objective truth-value.
Rather it offers its own, arguably superior, resources for expressing the content of mathematical statements in a precise and rigorous setting that has been specifically designed for this purpose.
So, the question about the provability or unprovability of any given mathematical statement becomes a sensible mathematical question.
Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories.
In addition, mathematics has a metalanguage; that is, names for mathematical statements and other parts of syntax, self-reference, proof and truth.
A mathematician friend recently explained that there is a whole class of mathematical statements that can be called independent.
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.
Mathematical platonism clearly motivates truth-value realism by providing an account of how mathematical statements get their truth-values.
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.
