In this essay I will show that whilst formalism is an attractive view it does not provide us with an adequate account of mathematics. I will begin with a brief outline of the basic position before going on to discuss it. Finally, I will discuss Hilbert’s programme.
In brief, formalism is the view that mathematics is the study of formal systems. This however does not tell the whole story and formalism can be divided into term formalism and game formalism (Shapiro, 2000: pp. 141-148). Term formalism is the view that mathematics is about characters or symbols. That is, the number 2 is just the character ‘2’. Whereas, game formalism is the view that mathematics is a game in the same way that chess is a game. There are characters, or pieces, that can only be manipulated according to specific rules. Consequently, mathematical practice is just like a game of chess and similarly meaningless.
On first glance, these views seem attractive for two reasons. First, it seems perfectly natural to agree that maths is just about symbol manipulation, what else could it be about? Second, formalism causes issues about the existence of numbers to fall away. Term formalism identifies numbers with characters and game formalism holds that mathematical symbols just are symbols.
There are, however problems with both these views. First, term formalism. If numbers are to be identified with characters then we encounter a problem. Consider the character ‘0’ and the character ‘0’. If the term formalist identifies numbers then since we have two separate characters we also have two separate numbers. A view I’m sure the term formalist does not want to defend. In which case the term formalist has to draw a distinction between token and type. For example, the word ‘formalism’ contains two occurrences of the letter ‘m’ or we could say that the word ‘formalism’ contains two tokens of type ‘m’. Tokens are physical objects but types are the abstract form of tokens. This is where the issue arises, in drawing this distinction the term formalist has moved away from concrete characters towards abstract objects and in doing so has destroyed the initial attraction. (Shapiro, 2000: pp. 142-143) The term formalist now needs to explain what these abstract types are. When considering game formalism another problem immediately arises. If mathematics is just a game then how do we account for its applicability? What is it that makes maths applicable but not chess? It could be objected here that chess does have it’s applications. For instance, chess can help with strategic thought or other such things. This however is not the same type of application. Mathematics has applications that directly relate to the external world. This is what the game formalist needs to account for.
Having highlighted the problems with game and term formalism I will now move onto discuss Hilbert’s programme. Hilbert’s aim in developing his programme was to establish the certainty of mathematics...