Formalism (mathematics) - Criticisms of Formalism

Criticisms of Formalism

Gödel indicated one of the weak points of formalism by addressing the question of consistency in axiomatic systems. More recent criticisms lie in the assertion of formalists that it is possible to computerize all of mathematics. These criticisms bring up the philosophical question of whether or not computers are able to think. Turing tests, named after Alan Turing, who created the test, are an attempt to provide criteria for judging when a computer is capable of thought. The existence of a computer which in principle could pass a Turing test would prove to formalists that computers will be able to do all of mathematics. However, there are opponents of this claim, such as John Searle, who came up with the "Chinese room" thought experiment. He presented the argument that while a computer may be able to manipulate the symbols that we give it, the machine could attach no meaning to these symbols. Since computers will not be able to deal with semantic content in mathematics (Penrose, 1989), they could not be said to "think."

Further, humans can create several ways to prove the same result, even if they might find it challenging to articulate such methods. Since creativity requires thought having a semantic foundation, a computer would not be able to create different methods of solving the same problem. Indeed, a formalist would not be able to say that these other ways of solving problems exist simply because they have not been formalized (Goodman, 1979).

Another critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent to the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.