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

‘there is no ontology here’ 377rf/(3)RuThis Fact in no way identifies the elements of Z/(3). It says how Z/(3) fitsinto the pattern of rings and ring morphisms. It implies there are exactly threeelements and they have specific algebraic relations to each other. It does not saywhat the elements are. Noether’s school specified them in various ways: as theintegers 0,1,2, or else as congruence classes of integers, or else as integers takenwith congruence modulo 3 as a new equality relation. Saunders Mac Lane contrastedthose last two approaches.⁸ Noether could have defined Z/(3) in somesuch way and proved the Fact from the definition; but in practice the Fact washer working definition. She knew it characterized Z/(3) up to isomorphism:Theorem 6. Suppose a ring Z/(3) and morphism r satisfy the Fact on Z/(3), as doanother Z/(3) ′ and r ′ . Then there is a unique ring isomorphism u : Z/(3) → Z(3) ′ suchthat ur = r ′ .rr′/(3)u/(3)′Proof. Since r ′ (3) = 0, the assumption on Z/(3) and r says there is a uniquemorphism u with ur = r ′ . Since r(3) = 0, the assumption on Z/(3) ′ and r ′says there is a unique v : Z/(3) ′ → Z/(3) with vr ′ = r. The composite vu is aring morphism with vur = vr ′ = r. Inotherwords,vur = 1 Z/(3) r, so uniquenessimplies vu = 1 Z/(3) . Similarly uv = 1 Z/(3) ′.She took the Fact as a case of the homomorphism theorem: a theorem or familyof theorems on quotient structures which she was rapidly expanding throughthe ten years up to her death.⁹ From the homomorphism theorem she deducedisomorphism theorems very much the way we deduced Theorem 6. Sherecastproblems of arithmetic as problems about morphisms, as for example problemsof arithmetic modulo 3 became problems about morphisms between Z/(3)and related structures.Equation 14.1 has solutions in other rings such as the quotient Z/(7) orthe ring Z[ √ 2] of numbers of the form a + b √ 2witha, b ordinary integers.⁸ See (Noether, 1927, n.6) and discussion in McLarty (2007b, Section5).⁹ She knew she had not yet measured the scope of this theorem (McLarty, 2006, Section4). Wecould call the present paper a case study of Noether’s homomorphism theorem.