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This article develops a critical investigation of the epistemological core of Hilbert’s foundational project, the so-called the finitary attitude. The investigation proceeds by distinguishing different senses of ‘number’ and ‘finitude’ that have been used in the philosophical arguments. The usual notion of modern pure mathematics, i.e. the sense of number which is implicit in the notion of an arbitrary finite sequence and iteration is one sense of number and finitude. Another sense, of older origin, is connected with practices of counting concrete things, and a third sense is linked up with the immediate intuitive experience of multitudes of concrete things. Hilbert’s finitism is examined with respect to these differences, and it will be shown that there is a tendency to conflate the different senses of number and finitude, a tendency which has been a source of problems in the discussion of the foundations of mathematics and in the philosophy of logic and language.

DOI: 10.1007/s11229-010-9823-2

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