**as a foundational principle for mathematics [eg p.5, p.158 of The Norton Library version] . As far as I can make out, Russell's basic argument contains these premises:**

*a priori*intuitions- Kant thought a priori intuitions were the foundation of mathematical belief [my own term].
- Kant's idea of belief was equivalent to Russell's idea of mathematical truth.
- Non-Euclidean geometry was discovered (after Kant) and proved that Euclidean propositions could
**not**be considered true, as they have alternatives. - THEREFORE: the foundation of mathematics cannot be a priori intuitions.

- Russell seems (at least briefly) to confuse mathematical truth with general truth. While mutually exclusive propositions from Euclidean and non-Euclidean geometry
*do*show the impossibility of such propositions having a general truth, there is no requirement that both propositions be believed simultaneously in the middle of a mathematical proof. - Russell seems, almost disingenuously, to be ignoring the viability of alternative a priori intuitions. I see no reason why an intuition of parellelism (in Euclid) cannot be replaced by an alternative intuition of -say- null parallelism (in spherical geometry). We can have either intuitions, as long as we do not try to hold both intuitions in mind simultaneously.
- Ultimately, everything Russell concludes is based on the fallacy that intuitions must be equivalent to general truth. By connecting these two things, insisting that they go together, and refuting 'truth', he pretends to refute 'intuitions'. This is called a
*straw man*argument.

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