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

358 colin mclartyTo put it generally, positions in one pattern or structure S 1 can be comparedto those in another S 2 by way of a function f : S 1 → S 2 .⁶ If f is one-to-one it isan injection and we say it identifies each position x ∈ S 1 with the value f (x) ∈ S 2 .This is a three place relation: x is identified with y via f ,orasanequationf (x) =y. The equation lies squarely in S 2 as f (x) and y are in S 2 . Some important injections,though, are taken as identity preserving. Whenanelementx ∈ S 1 is identifiedwithanelementy∈ S 2 via an identity-preserving injection S 1 ↣S 2 thenmathematicians will equate x with y. They state as a two place relation x = yand in this way they embed S 1 in S 2 . There is no definition of identity preserving.Mathematicians stipulate which injections are identity preserving only takingcare never to stipulate two distinct identity-preserving injections between thesame patterns. Nearly everyone takes R↣C as identity preserving.As a contrary example, mathematicians generally do not identify the complexnumber x 0 + x 1 i with the quaternionx 0 + x 1 i + 0j + 0kOne reason is that we could just as well identify it withx 0 + 0i + x 1 j + 0k or x 0 + 0i + 0j + x 1 k(Conway and Smith, 2003)or infinitely many other quaternions. All these injections preserve the usualstructure and are very useful and so no one is taken as identity preserving.In some contexts the software package Mathematica® does identify x 0 + x 1 iwith x 0 + x 1 i + 0j + 0k. So does the subject of crossed modules in grouprepresentation theory (Mac Lane, 2005, pp. 95f.). It is a matter of convention.Someday mathematicians may generally identify the complex numbers withsome quaternions. It will be decided by the demands of practice and not byprinciples of set theory.Mathematicians do not use the phrase identity preserving, Rather, they say forexample that ‘it is customary to identify’ certain elements of many differentalgebras with ‘the integers’ or the rational numbers, real numbers, etc., quiteapart from set theoretic definitions (Lang, 1993, p.90). Definitions are rarelygiven in Zermelo–Fraenkel set theoretic terms and when such terms are giventhey usually conflict with this custom. Yet the custom takes priority as withthe complex numbers. This obviates an historical problem MacBride sees forsome structuralists (2005,p.580). When two mathematicians posit ‘the’ natural⁶ See discussion on (MacBride, 2005, p.573), but we do not only use isomorphisms.