• DOI:10.1007/BF00485377
• Corpus ID: 46974744

Eudoxos and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics

@article{Stein2004EudoxosAD,
  title={Eudoxos and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics},
  author={Howard Stein},
  journal={Synthese},
  year={2004},
  volume={84},
  pages={163-211}
}

• H. Stein
• Published 1 August 1990
• Mathematics
• Synthese

According to Aristotle, the objects studied by mathematics have no independent existence, but are separated in thought from the substrate in which they exist, and treated as separable i.e., are "abstracted" by the mathematician. I In particular, numerical attributives or predicates (which answer the question 'how many?') have for "substrate" multitudes with a designated unit. 'How many pairs of socks?' has a different answer from 'how many socks?'. (Cf. Metaph. XIV i 1088a5ff.: "One la signifies that it is a measure of a multitude, and number lb that it is a measured multitude and a multitude of measures".) It is reasonable to see in this notion of a "measured multi tude" or a "multi tude of measures" just that of a (finite) set: the measures or units are what we should call the elements of the set; the requirement that such units be distinguished is precisely what differentiates a set from a mere accumulation or mass. There is perhaps some ambiguity in the quoted passage: the statement, "Number signifies that it is a measured multitude", might be taken either to identify numbers with finite sets, or to imply that the subjects numbers are predicated of are finite sets. Euclid's definition "a number is a multitude composed of units" points to the former reading (which implies, for example, that there are many two's a particular knife and fork being one of them). Number-words, on this interpretation, would be strictly construed as denoting infimae species of numbers. It is clearly in accord with this conception that Aristotle says, for example (in illustrating the "discreteness", as opposed to continuity, of number): "The parts of a number have no common boundary at which they join together. For example, if five is