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(Text as at 12/08/2007 10:17:46)
Systems of knowledge, such as mathematics, that are based on axioms and subsequent reasonings, do not constitute knowledge of the world, but may be used as models or pictures of the world.
- For example, as is well known, Euclidean geometry is an inadequate partial model of the world in the presence of matter (assuming the General Theory of Relativity to be true: in any case, the existence of this theory demonstrates that space is not necessarily Euclidean).
- Even ordinary arithmetic is not necessarily true of the world, as is shown by the existence of many abstract algebras. It is possible to imagine worlds in which the propositions of arithmetic did not hold (eg. addition could always be modulo 24, as the hours of a clock).
- In support of the contention that the rules of logic are necessarily true of the world, while those of arithmetic are not, it is sufficient to note that it has been proved, contra Whitehead and Russell, that Number theory cannot be reduced to logic. Godel's Incompleteness Theorem demonstrates (in more mathematical/logical language) that any axiomatisation of number theory contains theorems that can be stated but not proved within the formalism. Since mathematics (and, in particular, number theory) cannot be reduced to logic, it follows that any & all elements of a mathematical model may be false of the world while logic is true of it.
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