The three main branches in the philosophy of mathematics in the twentieth century are sketched in outline. The first of these, Logicism, has already been mentioned and was terminated abruptly by the work of Gödel (above). The second tendency appeared more radical: the formalists denied all meaning to mathematics and claimed that mathematics was only about symbols, rules for manipulating the symbols and the (symbolic) results which could be obtained by this manipulation. For them, questions about the truth or meaning of mathematical formulas were nonsensical. One could only say that a formula followed from certain other formulas using an agreed set of rules of derivation. Hilbert, the chief architect of Formalism, said
Still it is consistent with our finitary viewpoint to deny any meaning to logical symbols, just as we denied meaning to mathematical symbols, and to declare that the formulas of the logical calculus are ideal statements which mean nothing in themselves. [#!Hil83!#]
Formalists have no explanation of the tremendous power of mathematics applied to physical science. If mathematics has no meaning then why are the predictions of, say, quantum electrodynamics accurate to over twenty decimal places? Why is the correspondence between science and the world so good? Furthermore, however much the formalists deny any meaning to the symbols and formulas they use, in practice they appear to be highly influenced by the implications and meanings of their choice of language, axioms and inference rules. For the most part the language they adopt is not of random hieroglyphs but the language of arithmetic, set-theory and logic. And after the discovery of the paradoxes in set-theory the formalists too removed the axiom of comprehension from their axiom system.