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In Chapters 4 and 7 I argued that Poincaré's theory of predicativity is a consequence of his constructivist account of what is required in order to define a set. More generally, it is a consequence of his theory of meaning. The central component of Poincaré's theory of meaning is the notion of "verifiability in principle". This notion is, however, far from clear; and we must now enquire as to whether it is coherent. The notion of "verifiability in principle" is one of the cornerstones of Poincaré's philosophy of mathematics (and of science, in general). If either it or the more basic notion of "in principle possible" (for instance, in "constructible in principle", "provable in principle", "decidable in principle") cannot be regarded as possessing a clear, definite sense (as the strict finitist argues), then Poincaré's philosophy as a whole is in danger of losing all significance. The argument of this chapter is that Poincaré's weak "verificationist" theory of meaning is coherent, and when interpreted in the light of his background theory of the synthetic a priori, provides a stable position from which to fend off strict finitist attacks. Moreover, it provides a viable foundation for analysis.

DOI: 10.1007/978-1-349-22119-6_8

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