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Summary

‘Generality’ is an ambiguous and rather dangerous word, and we must be careful not to allow it to dominate our discussion too much. It is used in various senses both in mathematics and in writings about mathematics, and there is one of these in particular, on which logicians have very properly laid great stress, which is entirely irrelevant here. In this sense, which is quite easy to define, all mathematical theorems are equally and completely ‘general’.

‘The certainty of mathematics’, says Whitehead, ‘depends on its complete abstract generality.’ When we assert that 2 + 3 = 5, we are asserting a relation between three groups of ‘things’; and these ‘things’ are not apples or pennies, or things of any one particular sort or another, but just things, ‘any old things’. The meaning of the statement is entirely independent of the individualities of the members of the groups. All mathematical ‘objects’ or ‘entities’ or ‘relations’, such as ‘2’, ‘3’, ‘5’, ‘+’, or ‘=’, and all mathematical propositions in which they occur, are completely general in the sense of being completely abstract. Indeed one of Whitehead's words is superfluous, since generality, in this sense, is abstractness.