Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

what structuralism achieves 359numbers we need no criterion to tell if they posit the same ones. We can takethat as stipulated.13.1.2 Comparison with Resnik and ShapiroResnik tends to reject embeddings like R↣C, but he says we might justify thisone by ‘likening the historical development of the complex number systemto the step by step construction of a complicated pattern through addingpositions’ to the real number system (Resnik, 1997, p.215). He rightly offersthis as simile and not fact and so it is close to our appeal to stipulation. Overthe centuries some mathematicians took the complex numbers as including thereals, some took them as a different kind of thing, and some considered themnot genuine things.⁷ The history as a whole explains why today’s textbooksidentify R with a part of C. But the history is ambivalent and not widelyknown. The textbook stipulations are clear, overt, and authoritative.The textbook treatment of complex numbers might be analyzed into two stepsbased on Shapiro’s distinction of system and structure: First a ZF definition producesa system C of complex numbers, where each complex number is some specifiedZF set. Then abstraction forms a structure C of complex numbers ‘ignoringany features of them’ except their algebraic interrelations (Shapiro, 1997,p.73).⁸But then which are Lang’s complex numbers, or Conway and Smith’s?Lang defines a complex number as an ordered pair 〈x 0 , x 1 〉 where Conwayand Smith write x 0 + x 1 i (Lang, 2005,p.346). On the ‘system’ approach usingZF sets, Lang’s equation 〈x, 0〉 =x is literally false for any real number x.When Lang says to identify real numbers with the corresponding complexnumbers, is he revising his definition of complex numbers in some unspecifiedway? Or does he mean to treat some false equations as true? There are waysto do both but Lang never gives one. Conway and Smith are even moreambiguous as there is no standard definition of ‘formal expressions’ in ZF. Atany rate, if complex numbers are sets in a ‘system’ then ‘structures’ do notappear and at least for this quite typical case we would have to say mathematicsis not structuralist in practice.We can better understand complex numbers as places in a structure. Then‘x 0 + x 1 i’ and ‘〈x 0 , x 1 〉’ are not names of different sets. They are merelydifferent notations for the same position in the complex number structure.This explains why textbook authors routinely feel they can and must specifythat they identify each real number x with the complex x + 0i. Structural⁷ See the expert essay (Mazur, 2003) and the detailed exposition (Flament, 2003).⁸ ZF is the relevant set theory here. In categorical set theory each set is itself both ‘system’ and‘structure’ as its elements have no individuating features in the first place (McLarty, 1993).