On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others

Abstract

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative aspect of axiomatics for mathematical practice is brought to the fore.

From modules to lattices: Insight into the genesis of Dedekind'sDualgruppen

“…10 Dedekind's works on lattice theory did not attract as much attention as some of his other works. On the matter, one can see [Mehrtens, 1979], [Corry, 2004, 121-128] and [Schlimm, 2011]. The archival material on which this paper relies has not, as far as I am aware, been studied so far.…”

“…is called a Dualgruppe, if there are two operations ±, such that they create from two things α, β, two things α ± β, that are also in A and that satisfy [commutativity for + and -, associativity for + and -, and α ± (α ± β) = α (absorption)]. [Dedekind, 1897, 113, transl. in [Schlimm, 2011]].…”

“…They show that Dedekind's research process, here, implied many layers of computations that allowed him to explore and unfold module theory and the properties of its operations. 7 The computations and the stepwise generalization of the concept, however, largely disappeared from the published exposition of the theory, which is very general and abstract -or even axiomatic, as stated by [Schlimm, 2011].…”

Reliability of mathematical inference

“…Use algebraic abstraction. The modern algebraic method involves characterizing classes of structures of interest by specifying the axioms they satisfy and then reasoning abstractly about these structures via this axiomatic interface [58,59]. The fact that the algebraic method incorporates all three previous strategies helps explain its importance.…”

Dedekind and Hilbert on the Foundations of the Deductive Sciences

“…22 Recent philosophical work on Dedekind's ideal theory includes Tappenden (2005) and Avigad (2006). 23 Dedekind (1932c, p. 360, Was sind article 73): "Die Beziehungen oder Gezetze, welche ganz allein aus den Bedingungen α, β, γ , δ in 71 abgeleitet werden und deshalb in allen geordneten einfach unendlichen Systemen immer dieselben sind, wie auch die den einzelnen Elementen zufällig gegebenen Namen lauten mögen, bilden der nächsten Gegenstand der W i s s e n s c h a f t d e r Z a h l e n oder der A r i t h m e t i k." 24 See Schlimm (2011) for a recent philosophical discussion. 25 Dedekind (1932c, p. 113 might underlie this more abstract flavor of the dual group theory to explain the occurrences in Dedekind (1897) of the word 'law'.…”