Sinopsis
Two types of approach can be suggested to the question posed by the title of this chapter. On the one hand we might attempt to settle a priori on the criteria for mathematics and then review how far what we [ nd in di\ erent ancient cultures measures up to those criteria. Or we could proceed more empirically or inductively by studying those diverse traditions and then deriving an answer to our question on the basis of our [ ndings.
Both approaches are faced with di7 culties. On what basis can we decide on the essential characteristics of mathematics? If we thought, commonsensically, to appeal to a dictionary de[ nition, which dictionary are we to follow? 9 ere is far from perfect unanimity in what is on o\ er, nor can it be said that there are obvious, crystal clear, considerations that would enable us to adjudicate uncontroversially between divergent philosophies of mathematics. What mathematics is will be answered quite di\ erently by the Platonist, the constructivist, the intuitionist, the logicist, or the formalist (to name but some of the views on the twin fundamental questions of what mathematics studies, and what knowledge it produces). 9 e converse di7 culty that faces the second approach is that we have to have some prior idea of what is to count as ‘mathematics’ to be able to start our cross-cultural study. Other cultures have other terms and concepts and their interpretation poses delicate problems. Faced with evident divergence and heterogeneity, at what point do we have to say that we are not dealing with a di\ erent concept of mathematics, but rather with a concept that has nothing to do with mathematics at all? 9 e past provides ample examples of the dangers involved in legislating that certain practices and ideas fall beyond the boundaries of acceptable disciplines.
My own discussion here, which will concentrate largely on just two ancient mathematical traditions, namely Greek and Chinese, will owe more to the second than to the [ rst approach. Of course to study the ancient Greek or Chinese contributions in this area—their theories and their actual practices—we have to adopt a provisional idea of what can be construed as mathematical, principally how numbers and shapes or [ gures were conceived and manipulated. But as we explore further their ancient ideas of what the studies of such comprised, we can expect that our own understanding will be subject to modi[ cation as we proceed. We join up, as we shall see, with those problems in the philosophy of mathematics I mentioned: so in a sense a combination of both approaches is inevitable. Both the Greeks and the Chinese had terms for studies that deal, at least in part, with what we can easily recognize as mathematical matters, and this can provide an entry into the problems, though the lack of any exact equivalent to our notion in both cases is obvious from the outset. I shall [ rst discuss the issues as they relate to Greece before turning to the less familiar data from ancient China.
Content
- GEOGRAPHIES AND CULTURES
- Global
- Regional
- Local
- PEOPLE AND PRACTICES
- Lives
- Practices
- Presentation
- INTERACTIONS AND INTERPRETATIONS
- Intellectual
- Mathematical
- Historical
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