What is abstraction & what is it good for?
pp. 17-40
in: Irvine (ed), Physicalism in mathematics, Berlin, Springer, 1990Abstract
Being a nominalist convinced of the importance of mathematics is not a state to be recommended for its comfort. (Repeat several times a day: "The objects of mathematics are abstract. There are no abstract objects.") The oldest ingredient of many remedies aiming to reduce the cognitive tension goes back to Aristotle. Aphairesis refers to the way in which mathematicians may treat of objects which are inseparable from real things, by restricting consideration to only some aspects of sensible things while overlooking others. 1 This term was rendered by Boethius as abstractio, together (unfortunately) with chorismos ('separation"), which referred to the (for Aristotle fictitious) postulation of Platonic ideas.2 While the employment of abstraction to explain how knowledge of the general (universal) Arises of mathematical knowledge was there from the start.
