A philosophy of mathematics developed by L. E. J. Brouwer (1881–1966). Abstract objects such as numbers are not mind-independent objects, but are mental constructions. Consequently, it makes no sense to say that one exists unless one can actually construct it (mentally). It is therefore invalid to reason by reductio as follows: Suppose that $\neg \exists x\phi \left(x\right)$, and that this leads to a contradiction. Then $\exists x\phi \left(x\right)$. The principle of double negation and the principle of excluded middle are also invalid. For an intuitionist, much of classical mathematics has to be given up, though distinctive forms of intuitionist mathematics also become available. For example, in the theory of real numbers, as formulated intuitionistically, it can be shown that all real-valued functions are continuous (which is not true in classical mathematics). Later in the twentieth century intuitionist philosophy was generalized by Michael Dummett (1925–2011) to a view applicable, quite generally, to all language, based on the thought that meaning is determined by use, that is, is determined by assertability conditions.