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. Fo...
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...cond part of Gödel’s theorem shows that if T is consistent then the consistency of T cannot be proven within T. This leaves Hilbert’s programme in tatters. Hilbert had hoped to give a proof of the consistency of mathematics using finite methods but Gödel’s theorem shows that such a proof cannot be found. Hilbert’s programme cannot establish the certainty of mathematics. (Brown, 2008: pp. 76-82)
In conclusion, I have shown that the basic positions of formalism are unconvincing. I have also shown that whilst Hilbert’s programme failed for technical reasons there are also philosophical problems with it.
Brown, J., 2008. Philosophy of mathematics: a contemporary introduction to the world of proofs and pictures. 2nd Edition. London: Routledge.
Shapiro, S., 2000. Thinking about mathematics: the philosophy of mathematics. Oxford: Oxford University Press.
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