The Living Foundations of Mathematics

Abstract

Eric Livingston's book, The Ethnomethodological Foundations of Mathematics, addresses two of the most intriguing and difficult questions in the philosophy of mathematics. First, what is the nature of mathematical objects, for example, the circles and lines mentioned in geometrical theorems? Second, what is the nature of the compulsion that attaches to mathematical reasoning? The idealized and perfect circles whose properties are revealed in Euclid are a far cry from the poor, crooked things that a mathematics teacher draws on a blackboard. Again, the proofs of theorems seems to embody a peculiarly potent form of persuasion. How can we possess such superlative insights into such superlative things? Livingston says that the answer to these long-standing questions will be found by attending to certain of the local contingencies that surround particular episodes of mathematical reasoning and theoremproving. His aim is to get us to appreciate the moment-by-moment 'work' of mathematical reasoning at the 'mathematical work-site' (6). The claim running throughout his book is that by attending to these details we will come to understand how the upshot of this 'work' can be seen as possessing the transcendent qualities just described. Livingston says his concern is with the 'living foundations of mathematics' (x), in contrast to the 'classical' study of foundations. Classical studies, such as those of Russell, aim to provide a definition of mathematical concepts in terms of even more basic logical concepts. Their aim is to justify mathematics by deriving its operations, such as addition, from formalized logical operations of a kind that are even more primitive. However, all the fundamental processes of reasoning that underlie our